On the massiveness of the set of nonintegrable Hamiltonians

A problem concerning Hamiltonians that are reduced via a convergent Birkhoff transformation to a normal form is considered. There is a well-known classical result of C. L. Siegel on the massiveness of the set of nonintegrable Hamiltonians in a neighborhood of an equilibrium state of a system with $n$ degrees of freedom. The main result of this paper, Theorem 3, asserts that Siegel's theorem is valid for $n>2$ under an additional assumption on the Diophantine properties of the eigenvalues of the linearized system.