Pritula N. N.

This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.

We study finite-dimensional Moser-type reductions for the inverse nonlinear Korteweg–de Vries dynamical system and the Liouville integrability of these reductions in quadratures.

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.

We construct a hierarchy of Poisson Hamiltonian structures related to an “elliptic” spectral problem and determine the generating operators for the equation of asymmetric chiral 0 (3) — field.

Based on an analysis of a supersymmetric extension of the algebra of pseudodifferential operators on $ℝ^1$ an infinite hierarchy of supersymmetric Lax-integrable nonlinear dynamical systems is constructed by means of the Yang-Baxter $ℛ$-equation method. The structure of these systems on reduced invariant submanifolds specified by a natural invariant Lax-type spectral problem is investigated.