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We develop an approach analogous to classical Frobenius theory for the analysis of singularities of ODEs in the case of discrete dynamical systems. Our methodology is based on the Roman-Rota theory of finite operators and relies crucially on the idea of preserving the Leibniz rule on a lattice of points by means of the notion of Rota algebras. The relevant cases of the Bessel, Hermite and Airy equations are discussed., Comment: 19 pages, 1 figure

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically; besides, the underlying St\"ackel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field., Comment: 25 pages, no figures. Includes minor revisions enhancing clarity suggested by referees for Proceedings of Royal Society A and some additional comments

This is a review of group entropy and its application to permutation complexity. Specifically we revisit a new approach to the notion of complexity in time serie analysis, based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviours., Comment: 24 pages; 4 figures

We report the Astrorivelatore Gamma ad Immagini LEggero (AGILE) observations of solar flares, detected by the on board anticoincidence system in the 80-200 keV energy range, from 2007 May 1st to 2022 August 31st. In more than 15 yr, AGILE detected 5003 X-ray, minute-lasting transients, compatible with a solar origin. A cross-correlation of these transients with the Geostationary Operational Environmental Satellites (GOES) official solar flare database allowed to associate an intensity class (i.e., B, C, M, or X) to 3572 of them, for which we investigated the main temporal and intensity parameters. The AGILE data clearly revealed the solar activity covering the last stages of the 23rd cycle, the whole 24th cycle, and the beginning of the current 25th cycle. In order to compare our results with other space missions operating in the high-energy range, we also analyzed the public lists of solar flares reported by RHESSI and Fermi Gamma-ray Burst Monitor. This catalog reports 1424 events not contained in the GOES official dataset, which, after statistical comparisons, are compatible with low-intensity, short-duration solar flares. Besides providing a further dataset of solar flares detected in the hard X-ray range, this study allowed to point out two main features: a longer persistence of the decay phase in the high-energy regime, with respect to the soft X-rays, and a tendency of the flare maximum to be reached earlier in the soft X-rays with respect to the hard X-rays. Both these aspects support a two-phase acceleration mechanism of electrons in the solar atmosphere., Comment: 22 pages, 10 figures

We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable., Comment: 9 pages, no figures

In this work, we introduce the notion of polarization of generalized Nijenhuis torsions and establish several algebraic identities. We prove that these polarizations are relevant in the characterization of Haantjes $C^{\infty}$(M)-modules of operator fields., Comment: 13 pages, no figures

We report the AGILE observations of GRB 220101A, which took place at the beginning of 1st January 2022 and was recognized as one of the most energetic gamma-ray bursts (GRBs) ever detected since their discovery. The AGILE satellite acquired interesting data concerning the prompt phase of this burst, providing an overall temporal and spectral description of the event in a wide energy range, from tens of keV to tens of MeV. Dividing the prompt emission into three main intervals, we notice an interesting spectral evolution, featuring a notable hardening of the spectrum in the central part of the burst. The average fluxes encountered in the different time intervals are relatively moderate, with respect to those of other remarkable bursts, and the overall fluence exhibits a quite ordinary value among the GRBs detected by MCAL. However, GRB 220101A is the second farthest event detected by AGILE, and the burst with the highest isotropic equivalent energy of the whole MCAL GRB sample, releasing E_iso=2.54x10^54 erg and exhibiting an isotropic luminosity of L_iso=2.34x10^52 erg/s (both in the 400 keV - 10 MeV energy range). We also analyzed the first 10^6 s of the afterglow phase, using the publicly available Swift XRT data, carrying out a theoretical analysis of the afterglow, based on the forward shock model. We notice that GRB 220101A is with high probability surrounded with a wind-like density medium, and that the energy carried by the initial shock shall be a fraction of the total E_iso, presumably near 50%., Comment: 16 pages, 9 figures, 4 tables

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We propose a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level $l$ vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras., Comment: 20 pages, no figures

The complexity of browsers has steadily increased over the years, driven by the continuous introduction and update of Web platform components, such as novel Web APIs and security mechanisms. Their specifications are manually reviewed by experts to identify potential security issues. However, this process has proved to be error-prone due to the extensiveness of modern browser specifications and the interplay between new and existing Web platform components. To tackle this problem, we developed WebSpec, the first formal security framework for the analysis of browser security mechanisms, which enables both the automatic discovery of logical flaws and the development of machine-checked security proofs. WebSpec, in particular, includes a comprehensive semantic model of the browser in the Coq proof assistant, a formalization in this model of ten Web security invariants, and a toolchain turning the Coq model and the Web invariants into SMT-lib formulas to enable model checking with the Z3 theorem prover. If a violation is found, the toolchain automatically generates executable tests corresponding to the discovered attack trace, which is validated across major browsers. We showcase the effectiveness of WebSpec by discovering two new logical flaws caused by the interaction of different browser mechanisms and by identifying three previously discovered logical flaws in the current Web platform, as well as five in old versions. Finally, we show how WebSpec can aid the verification of our proposed changes to amend the reported inconsistencies affecting the current Web platform., Comment: Submitted to IEEE S&P '23 on 19 Aug 2022

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this \textquotedblleft ordinal\textquotedblright\ methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of visible permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of visible permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes., Comment: 22 pages, 4 figures

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($\omega, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathscr{H}$, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many $\omega\mathscr{H}$ structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures., Comment: 37 pages, no figures

Related-domain attackers control a sibling domain of their target web application, e.g., as the result of a subdomain takeover. Despite their additional power over traditional web attackers, related-domain attackers received only limited attention by the research community. In this paper we define and quantify for the first time the threats that related-domain attackers pose to web application security. In particular, we first clarify the capabilities that related-domain attackers can acquire through different attack vectors, showing that different instances of the related-domain attacker concept are worth attention. We then study how these capabilities can be abused to compromise web application security by focusing on different angles, including: cookies, CSP, CORS, postMessage and domain relaxation. By building on this framework, we report on a large-scale security measurement on the top 50k domains from the Tranco list that led to the discovery of vulnerabilities in 887 sites, where we quantified the threats posed by related-domain attackers to popular web applications., Comment: Submitted to USENIX Security '21 on 16 Oct 2020

Permutation entropy measures the complexity of deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or just permutations. The reasons for the increasing popularity of this entropy in time series analysis include that (i) it converges to the Kolmogorov-Sinai entropy of the underlying dynamics in the limit of ever longer permutations, and (ii) its computation dispenses with generating and ad hoc partitions. However, permutation entropy diverges when the number of allowed permutations grows super-exponentially with their length, as is usually the case when time series are output by random processes. In this Letter we propose a generalized permutation entropy that is finite for random processes, including discrete-time dynamical systems with observational or dynamical noise., Comment: 9 pages, 5 figures

Session management is a fundamental component of web applications: despite the apparent simplicity, correctly implementing web sessions is extremely tricky, as witnessed by the large number of existing attacks. This motivated the design of formal methods to rigorously reason about web session security which, however, are not supported at present by suitable automated verification techniques. In this paper we introduce the first security type system that enforces session security on a core model of web applications, focusing in particular on server-side code. We showcase the expressiveness of our type system by analyzing the session management logic of HotCRP, Moodle, and phpMyAdmin, unveiling novel security flaws that have been acknowledged by software developers.

We define the class of multivariate group entropies as a novel set of information - theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality class of complex systems; in particular, we introduce a general entropy, representing a suitable information measure for this class. We also show that the group-theoretical structure associated with our multivariate entropies can be used to define a large family of exactly solvable discrete dynamical models. The natural mathematical framework allowing us to formulate this correspondence is offered by the theory of formal groups and rings., Comment: 16 pages

We present a mathematical construction of new quantum information measures that generalize the notion of logarithmic negativity. Our approach is based on formal group theory. We shall prove that this family of generalized negativity functions, due their algebraic properties, is suitable for studying entanglement in many-body systems. Under mild hypotheses, the new measures are computable entanglement monotones. Also, they are composable: their evaluation over tensor products can be entirely computed in terms of the evaluations over each factor, by means of a specific group law. In principle, they might be useful to study separability and (in a future perspective) criticality of mixed states, complementing the role of R\'enyi's entanglement entropy in the discrimination of conformal sectors for pure states., Comment: 18 pages; to appear in Quantum Information Processing, 2021

We introduce a class of information measures based on group entropies, allowing us to describe the information-theoretical properties of complex systems. These entropic measures are nonadditive, and are mathematically deduced from a series of natural requirements. In particular, we introduce an extensivity postulate as a natural requirement for an information measure to be meaningful. The information measures proposed are suitably defined for describing universality classes of complex systems, each characterized by a specific phase space growth rate function., Comment: 14 pages. No figures

A notion of one-dimensional formal ring is presented. It consists of a triple $(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket x,y\rrbracket}$, the first one defining a one-dimensional formal group law over $A$ and the second one providing a second composition law satisfying axiomatic properties of compatibility with the first one. For a characteristic-zero ring $A$, a large class of one-dimensional formal rings can be obtained by constructing a new composition law, defined in terms of the group logarithm associated with a given formal group law $\Phi(x,y)$, and associative and distributive with respect to it. A natural $n$-dimensional generalization of the previous construction is also proposed; curves on $n$-dimensional formal rings are introduced. The higher-dimensional theory allows us to define a generalization of the ring $W(A)$ of Witt vectors over a ring $A$, which is recovered by means of a specific choice of the associated group logarithm. The composition laws of our generalized Witt ring are defined in terms of an underlying formal ring structure. Examples of formal rings related to Hirzebruch's theory of genera are explicitly computed. Finally, we also propose the examples of Euler's and Abel's formal rings., Comment: 20 pages

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume $W$ on $N$. We review the ensuing entropies, discuss the composability axiom, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective., Comment: 12 pages - invited contribution to the journal Entropy's special issue "Nonadditive Entropies and Complex Systems"

We propose a new, infinite class of brackets generalizing the Fr\"olicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form., Comment: 24 pages; some results added

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for a large class of entropies. In addition, a method for defining new entropies, using previously known ones with some desired group-theoretical properties is proposed. In the second part of this work, the information geometrical counterpart of the previous construction is examined and a general class of divergences are proposed and studied. Finally, a method of constructing new divergences from known ones is discussed; in particular, some results concerning the Riemannian structure associated with the class of divergences under investigation are formulated., Comment: 17 pages

We present WPSE, a browser-side security monitor for web protocols designed to ensure compliance with the intended protocol flow, as well as confidentiality and integrity properties of messages. We formally prove that WPSE is expressive enough to protect web applications from a wide range of protocol implementation bugs and web attacks. We discuss concrete examples of attacks which can be prevented by WPSE on OAuth 2.0 and SAML 2.0, including a novel attack on the Google implementation of SAML 2.0 which we discovered by formalizing the protocol specification in WPSE. Moreover, we use WPSE to carry out an extensive experimental evaluation of OAuth 2.0 in the wild. Out of 90 tested websites, we identify security flaws in 55 websites (61.1%), including new critical vulnerabilities introduced by tracking libraries such as Facebook Pixel, all of which fixable by WPSE. Finally, we show that WPSE works flawlessly on 83 websites (92.2%), with the 7 compatibility issues being caused by custom implementations deviating from the OAuth 2.0 specification, one of which introducing a critical vulnerability.

We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form., Comment: 29 pages, no figures

The requirement that an entropy function be composable is key: it means that the entropy of a compound system can be calculated in terms of the entropy of its independent components. We prove that, under mild regularity assumptions, the only composable generalized entropy in trace form is the Tsallis one-parameter family (which contains Boltzmann-Gibbs as a particular case). This result leads to the use of generalized entropies that are not of trace form, such as R\'enyi's entropy, in the study of complex systems. In this direction, we also present a characterization theorem for a large class of composable non-trace-form entropy functions with features akin to those of R\'enyi's entropy., Comment: 15 pages, no figures

The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set $A$ of sites of the subsystem considered and the set $K$ of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets $A$ and $K$ to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets $A$ and $K$ consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and $r$-partite information generalizing the known ones for the single block case., Comment: 12 pages, 5 figures, typeset in LaTeX

We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear forms is also proposed. We consider both polynomial deformations and a class of transformations related to the action of linear operators on the entries of a given bilinear form. Transformation formulae among new and old polynomial sequences are determined. Finally, integrable hierarchies of evolution equations arising from the factorization of a time deformation of the moment matrix are presented., Comment: 52 pages

The volume of phase space may grow super-exponentially ("explosively") with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create new emergent states. Standard ensemble theory can break down as we demonstrate in a simple model reminiscent of complex systems where new collective states emerge. We present an axiomatically defined entropy and argue that it is extensive in the micro-canonical, equal probability, and canonical (max-entropy) ensemble for super-exponentially growing phase spaces. This entropy may be useful in determining probability measures in analogy with how statistical mechanics establishes statistical ensembles by maximising entropy., Comment: 18 pages, 1 figure. This version is significantly expanded and improved as a result of referee comments. This is the manuscript accepted for publication in J Phys A

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$ for $a, b, c$ smooth real functions defined on an open set of $\mathbb{R}^2$. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the appropriate way to define the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts algebraic methods from invariant theory for representations of compact Lie groups on the space of quadratic forms on $\mathbb{R}^n$, $n \geq 2$. With that we obtain an algorithm to compute general expressions of quadratic forms. Now, symmetric quadratic 1-forms are in one-to-one corrspondence with equivariant quadratic forms on the plane, so these are treated here as a particular case. We then apply the result to obtain the general forms of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group $\mathbf{O}(2)$.

We introduce a general class of su$(1|1)$ supersymmetric spin chains with long-range interactions which includes as particular cases the su$(1|1)$ Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su$(1|1)$ permutation operator, and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low energy excitations and the low temperature behavior of the free energy, which coincides with that of a $(1+1)$-dimensional conformal field theory (CFT) with central charge $c=1$ when the chemical potential lies in the critical interval $(0,\mathcal E(\pi))$, $\mathcal E(p)$ being the dispersion relation. We also analyze the von Neumann and R\'enyi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson $(1+1)$-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge $c=1$. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su$(1|1)$ elliptic chain., Comment: 13 pages, 6 figures, typeset in REVTeX

We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts., Comment: 5 pages

We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models with su$(m+1)$ spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su$(m+1)$ type. We evaluate in closed form the reduced density matrix of a block of $L$ spins when the whole system is in its ground state, and study the corresponding von Neumann and R\'enyi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as $a\log L$ when $L$ tends to infinity, where the coefficient $a$ is equal to $(m-k)/2$ in the ground state phase with $k$ vanishing su$(m+1)$ magnon densities. In particular, our results show that none of these generalized Lipkin-Meshkov-Glick models are critical, since when $L\to\infty$ their R\'enyi entropy $R_q$ becomes independent of the parameter $q$. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su$(m+1)$ Lipkin-Meshkov-Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when $m-k\ge3$. Finally, in the su$(3)$ case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su$(3)$. This is also true in the su$(m+1)$ case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an $(m+1)$-simplex in $\mathbf R^m$ whose vertices are the weights of the fundamental representation of su$(m+1)$., Comment: Typeset with LaTeX, 32 pages, 3 figures. Final version with corrections and additional references

In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized St\"ackel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As an application, we recover the Haantjes manifolds for the rational Calogero model with three particles and for the Benenti systems., Comment: We apply the theory of symplectic-Haantjes manifolds introduced in arXiv:1405.5118 (which has been merged with arxiv:1508.04629) to the class of generalized St\"ackel systems

We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. This structure, expressed in terms of suitable operators whose Haantjes torsion vanishes, encodes the main features of the notion of integrability, and in particular, under certain hypotheses, allows to solve the problem of determining separation of variables for a given system in an algorithmic way. As an application of the theory, we prove theorems ensuring the existence of a large class of completely integrable systems in the Euclidean plane, constructed starting from a prescribed Haantjes structure. At the same time, we also show that some of the most classical examples of Hamiltonian systems in n dimensions, as for instance the Gantmacher and St\"ackel classes, all possess a natural Haantjes structure., Comment: This manuscript has been merged with the paper "Haantjes manifolds of classical integrable systems", arXiv:1405.5118