Classification of geodesic flows of Liouville metrics on the two-dimensional torus up to topological equivalence

The basic results of the theory of A. T. Fomenko on the topological properties of integrable Hamiltonian systems with two degrees of freedom are used to obtain the topological classification of geodesic flows on the torus $T^2$ with a Bott integral that is quadratic in the impulses, to state a criterion for a system to be a Bott system in terms of the function of the metric on $T^2$, to explicitly calculate the Fomenko invariant $W$ (an untagged molecule) and the Fomenko–Zieschang invariant $W^*$ (atagged molecule), and to completely describe the place occupied by the systems under consideration in the molecular table of complexity.