I have the Hamiltonian of some dynamical system:
$$ H =\frac{p_1^2+p_2^2}{u(x_1)+v(x_2)}$$
Coordinates x=(x1, x2) change on the torus, therefore the functions u and v are 2pi-periodic. That is, the phase space is the cotangent bundle of a two-dimensional torus
I found another first integral:
$$ f = \frac{p_2^2u(x_1)-p_1^2v(x_2)}{u(x_1)+v(x_2)} $$
According to the Arnold-Liouville theorem, the joint level set of the Hamiltonian and the first integral is something diffeomorphic to a two-dimensional torus or cylinder.
My question is how to describe and study this joint level set in order to explain what the Louisville tori look like.
I got to the point where this set is described by these two functions: $$ f(x_1, p_1)=p_1^2-c_1u(x_1)+c_2 = 0 $$ $$ f(x_2, p_2)=p_2^2-c_1v(x_2)-c_2 = 0 $$
And now I don't understand what to do next.