# Hentosh О. Ye.

### Differential-geometric structure and the Lax – Sato integrability of a class of dispersionless heavenly type equations

Hentosh О. Ye., Prikarpatskii Ya. A., Pritula N. N.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 293-297

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.

### Lie-algebraic structure of the Lax-integrable (2| 1+ 1) -dimensional supersymmetric matrix dynamical systems

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1310-1323

By using a specially constructed Backlund transformation, we obtain the Hamiltonian representation for the hierarchy of Laxtype flows on the dual space to the Lie algebra of matrix superintegral-differential operators with one anticommutative variable, coupled with suitable evolutions of eigenfunctions and adjoint eigenfunctions of the associated spectral problems. We also propose the Hamiltonian description of the corresponding set of the hierarchies of additional homogeneous symmetries (squared eigenfunction symmetries). The connection between these hierarchies and the Lax-integrable (2| 1+1)-dimensional supersymmetric matrix nonlinear dynamical systems and their triple Lax-type linearizations is analyzed.

### Lax-integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional reduction of Neumann type

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 906-921

A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy, we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting vector fields generated by the hierarchy.

### Compatibly bi-Hamiltonian superconformal analogs of Lax-integrable nonlinear dynamical systems

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 887–900

Compatibly bi-Hamiltonian superanalogs of the known Lax-integrable nonlinear dynamical systems are obtained by using a relation for the Casimir functionals of central extensions of the Lie algebra of superconformal even vector fields and its adjoint semidirect sum.

### Lie-algebraic structure of (2 + 1)-dimensional Lax-type integrable nonlinear dynamical systems

Hentosh О. Ye., Prikarpatskii A. K.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 939–946

A Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems is obtained via some special Båcklund transformation. The connection of this hierarchy with Lax-integrable two-metrizable systems is studied.