Polynomial integrals of mechanical systems on a torus with a singular potential

Abstract

The problem on integrability of the equations of motion of a material point on an n-dimensional Euclidean torus under the action of a force field with the potential energy having singularities at a finite number of points is considered. It is assumed that these singularities contain logarithmic coefficients and, consequently, have a more general form in comparison with power features. The potentials having power-type singularities were considered previously by V.V. Kozlov and D.V. Treshchev. In this work, it is proved that the equations of motion in the problem under consideration admit no nontrivial momentum-polynomial first integral with integrable coefficients on this torus.