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The symmetry approach to the classification of evolution integrable partial differential equations (see, for example~\cite{MikShaSok91}) produces an infinite series of functions, defined in terms of the right hand side, that are conserved densities of any equation having infinitely many infinitesimal symmetries. For instance, the function $\frac{\partial f}{\partial u_{x}}$ has to be a conserved density of any integrable equation of the~KdV type~$u_t=u_{xxx}+f(u,u_x)$. This fact imposes very strong conditions on the form of the function~$f$. In this paper we construct similar canonical densities for equations of the Boussinesq type. In order to do that, we write the equations as evolution systems and generalise the formal diagonalisation procedure proposed in \cite{MSY} to these systems.

We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further describe non-abelian integrable equations, especially matrix (ODE and PDE) systems. Some non-evolutionary integrable equations are studied using a formulation of formal recursion operators that allows to study non-diagonalisable systems of evolution equations., Comment: To be published in: Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 2, N. Euler and M.C. Nucci (editors), CRC Press, Boca Raton, 2019. arXiv admin note: text overlap with arXiv:1711.10624

We study integrability --in the sense of admitting recursion operators-- of two nonlinear equations which are known to possess compacton solutions: the $K(m,n)$ equation introduced by Rosenau and Hyman \[ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , \] and the $CSS$ equation introduced by Coooper, Shepard, and Sodano, \[ D_t(u) + u^{l-2}D_x(u) + \alpha p D_x (u^{p-1} u_x^2) + 2\alpha D_x^2(u^p u_x) = 0 \; . \] We obtain a full classification of {\em integrable $K(m,n)$ and $CSS$ equations}; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of $CSS$., Comment: 24 pages

We derive the general structure of the space of formal recursion operators of nonevolutionary equations~$q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})$. This allows us to classify integrable Lagrangian systems with a higher order Lagrangian of the form~$\mathscr{L}=\frac12 L_2(q_{xx}, q_x, q)\,q_t^2 + L_1(q_{xx}, q_x, q)\, q_{t} + L_0(q_{xx}, q_x, q)$. The key technique relays on exploiting a homogeneity of the determining equations of formal recursion operators. This technique allows us to extend the main results to more general equations~$q_{tt}=f(q,q_{x},\ldots,q_{n};q_{t},q_{xt},\ldots,q_{mt})$.

We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of the nonlinear Schrodinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrodinger equation. On the other hand, if we start with a non-integrable lattice equation we may obtain a non-integrable NLS equation. This conjecture is confirmed by many examples., Comment: 12 pages

We give an analytic, sufficient condition for the existence of the Backlund transformation between the semiinfinite Toda and Volterra lattices, in the complex case, extending previous results given for the real case., Comment: 7 pages