The non-singular surfaces of constant energy associated to the integrable Hamiltonian systems defined on four-dimensional symplectic manifolds are studied. It is shown that the irreducible surfaces of constant energy corresponding to such integrable systems are completely defined by their fundamental groups. For the Hamiltonian systems it is proven that the surfaces of constant energy are irreducible.
It is established that all closed periodic solutions of the Hamiltonian systems are non-homotopical null and the circles corresponding to the maximum or minimum values of the Bott integral are non-homotopical to each other. It is pointed out that the results on topological integrability are important in the study of Liouville torus bifurcations.