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We present a reduction theory for first order Lagrangian field theories which takes into account the conservation of momenta. The relation between the solutions of the original problem with a prescribed value of the momentum and the solutions of the reduced problem is established. An illustrative example is discussed in detail., Comment: v2: a few references added, some typos corrected

We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the $\tau $-representation. We show that in this stage a natural Ad-invariant bilinear form does exist, allowing for a straightforward application of the AKS theory, and giving rise to Manin triple structure, thus bringing the problem to the realm of Lie bialgebras and Poisson-Lie groups.

We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems., Comment: 46 pages

In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems.

We study the deep connection between integrable models and Poisson-Lie T-duality working on a finite dimensional example constructed on SL(2,C) and its Iwasawa factors SU(2) and B. We shown the way in which Adler-Kostant-Symes theory and collective dynamics combine to solve the equivalent systems from solving the factorization problem of an exponential curve in SL(2,C). It is shown that the Toda system embraces the dynamics of the systems on SU(2) and B., Comment: 34 pages

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the causative agent of coronavirus disease 2019 (COVID-19). Whether SARS-CoV-2 can trigger an autoimmune reaction against platelets and red blood cells remains unclear. Herein, we report a case of COVID-19 pneumonia associated with severe immune thrombocytopenia and hemolytic anemia. An 83-year-old woman was admitted to the hospital because of both dyspnea and diffuse mucocutaneous bleeding. Exams revealed hemolytic anemia (HA), severe immune thrombocytopenia (ITP), and bilateral pneumonia. Molecular testing confirmed a diagnosis of COVID-19 pneumonia. Thrombocytopenia did not respond to first-line treatment with immunoglobulin, corticosteroids, and platelet transfusions. Addition to therapy of the thrombopoietin receptor agonist, eltrombopag, resulted in full recovery. COVID-19 can be associated with ITP and HA. There are neither guidelines nor clinical experience on the treatment of COVID-19-associated ITP and our case, showing complete response to eltrombopag, may help clinicians in their practice during the COVID-19 pandemic., Plain Language Summary: The case of an 83-year-old woman with COVID-19 pneumonia associated with two severe blood diseases that cause platelet and red cell destruction Coronavirus disease 2019 (COVID-19) is caused by a virus called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). We do not know exactly whether this virus can stimulate our immune system to react against platelets and red blood cells. Herein, we report a case of COVID-19 pneumonia associated with two severe blood diseases, immune thrombocytopenia, which causes platelet destruction, and hemolytic anemia, which causes red cell destruction. An 83-year-old woman was admitted to the hospital because of both difficulty in breathing and diffuse bleeding in mucosae and skin. Exams revealed hemolytic anemia, severe immune thrombocytopenia, and pneumonia in both lungs. Molecular testing confirmed a diagnosis of COVID-19 pneumonia. The first treatment with immunoglobulin, corticosteroids, and platelet transfusions was not enough to cure thrombocytopenia; the addition of eltrombopag which acts on the thrombopoietin receptor agonist resulted in full recovery. COVID-19 can be present together with immune thrombocytopenia and hemolytic anemia. As there are no guidelines on the treatment of immune thrombocytopenia in patients with COVID-19 and the clinical experience is limited, the complete response achieved with eltrombopag may help clinicians in their practice during the COVID-19 pandemic., Competing Interests: Conflict of interest statement: The authors declare that there is no conflict of interest., (© The Author(s), 2021.)