# doc. RNDr. Artur Sergyeyev, Ph.D.

## Main Scientific Results

- Proof of maximal superintegrability for a large class of Benenti systems (natural Hamiltonian dynamical systems admitting a special conformal Killing tensor) - joint work with M. Blaszak (see [A.3]).
- For a large class of (1+1)-dimensional evolution systems, new easily verifiable sufficient conditions ensuring the locality of symmetries generated via the repeated application of recursion operator (see [A.2, A.6, B.1]) or repeated commutation with master symmetry (see [A.9, B.1]) were obtained. In particular, using these results we proved the Maltsev--Novikov conjecture on weak nonlocality of higher recursion operators, Hamiltonian and symplectic structures of integrable systems in (1+1) dimensions (see [A.2]).
- A new description of Hamiltonian operators compatible with a given
*nondegenerate* Hamiltonian operator, applicable for both finite- and infinite-dimensional case (see [A.5]).
- The complete classification of (1+1)-dimensional evolution systems admitting a generalized conditional symmetry (GCS) of prescribed form, including (unlike the known results of Svirshchevskii, Kamran, Milson, and Olver and Doyle) the case of explicitly time-dependent GCS (see [A.4, A.10]).
- All local generalized symmetries of the Bakirov system, including those explicitly dependent on independent variables, were found. In particular, it was shown that this system has only one non-Lie-point local generalized symmetry. This result, generalizing earlier work of Beukers, Sanders, and Wang, completes the refutation of the popular conjecture that the existence of one non-Lie point generalized symmetry for a system of PDEs implies the existence of infinitely many such symmetries (see [A.12]).
- An explicit formula for the leading term of commutator of two generalized symmetries of sufficiently high order was obtained (see [A.11]). This formula is valid for a large class of (1+1)-dimensional evolution systems with constraints. It possesses a number of useful applications and provides the complete solution of A.M. Vinogradov's problem of 'evaluation from the top' for the symmetry algebras of these systems.
- The first example of parasupersymmetric relativistic quantum-mechanical system with non-oscillator-like interaction was found. The complete set of eigenvalues and eigenvectors belonging to the discrete energy spectrum for this model was found (see [C.5]).