Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees

We consider systems with toric configuration space and kinetic energy in the form of a "flat" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by "natural" Hamiltonian systems of differential equations. If $V$ is replaced by $\varepsilon V$, where $\varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $\varepsilon$ is a part of the "main problem of dynamics" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of "higher" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.