On genericity of integrable Hamiltonian systems of Bott type

Integrable Hamiltonian systems on a symplectic manifold are considered in this paper. A special role among them is played by the so-called systems of Bott type on a given energy level.
Numerous investigations of concrete systems of classical mechanics and mathematical physics have shown that the systems are of Bott type for almost all energy levels. Therefore, an important question arises: is the property of being of Bott type in some sense a generic property? This paper is devoted to studying this question. The main result is that the set of systems of Bott type is of first category in the set of all integrable systems (in the weak metric) (Theorems 2.1 and 2.4). Recall that a set of first category is one that can be represented as a countable union of nowhere dense sets. However, this set is dense in the set of systems satisfying an additional condition (Theorem 3.1). The set of systems of Bott type is open in the strong metric (Theorem 4.1).