Methods for constructing invariant cubature formulasfor integrals over the surface of a torus in ${\mathbb R}^3$

The article considers the question of constructing cubature formulas for the surface of a torus $T$ in ${\mathbb R}^3$, invariant under the group $G$ generated by reflections of $T$ into itself. For currently known invariant cubature formulas with a degree of accuracy greater than $3$, the number of nodes significantly exceeds the minimum possible. The article proposes invariant cubature formulas of degree $5$ and $7$ for the surface of a torus with a number of nodes close to the minimum. Tables of values of nodes and coefficients of the constructed cubature formulas are given. The dependence of these values on the ratio of the radii of the guide and generatrix of the torus circles is studied. For construction, the method of invariant cubature formulas is used, based on the theorem of S. L. Sobolev.