Limits of the Formal Integrals of Motion

We consider a formal (approximate) integral of motion in Hamiltonians of the form $H=\frac{1}{2}(X^2+Y^2+\omega_1^2x^2+\omega_2^2y^2)+\epsilon(\eta xy^2+\alpha x^3+\beta x^2y+\gamma y^3)$ generalizing previous cases with $\beta=\gamma=0$. First we give the general form of this integral when $\omega_1/\omega_2$ is irrational and then we consider the case of commensurable frequencies. In particular we study the integrals for the resonances $\omega_1/\omega_2=4/1, 5/1, 3/2, 4/3, 3/1$ and $2/1$. We also calculate the invariant curves and the orbits in the cases $\omega_1/\omega_2=2/1$ and $1/1$ (with $\beta=\gamma=0$) and we compare the exact-numerical and the theoretical results predicted by the formal integral when $\beta\gamma\neq0$. In the special case $\omega_1/\omega_2=1/1$ we find an integral when $\beta=\gamma=0$ and $\eta\alpha\neq0$ or $\eta=\alpha=0$ and $\beta\gamma\neq 0$, but this is not possible when $\eta\alpha\beta\gamma\neq 0$. However, we find that the invariant curves and the orbits can be approximated by a non-resonant integral with $\omega_1/\omega_2=5\sqrt{2}/7=1.010\dots$.