On the density of nonintegrable Hamiltonian systems close to being billiard (Q1605612)

The author considers Hamiltonian systems with the Hamiltonian \(H=\frac 12(p_1^2+p_2^2)+V(f(x_1,x_2))\), where \(f(x_1,x_2)\) is a polynomial and the equation \(f(x_1,x_2)=0\) determines a closed curve in \(\mathbb{R}^2\) which bounds a certain region. If \(V\) is a meromorphic function in a neighborhood of zero that has a pole at zero, then we have a one-parameter family of Hamiltonian systems that is close to the billiard system. It is proved that the nonintegrable Hamiltonian systems from this class form an everywhere dense subset. It is an analogue of the famous Siegel theorem [see \textit{C. L. Siegel}, Math. Ann. 128, 144-170 (1954; Zbl 0057.32002)].