Consider the system $$ \begin{aligned} & \dot{q}_1=q_1 \cos(p_1-p_2) \\ & \dot{q}_2=-q_1 \cos(p_1-p_2) \\ & \dot{p}_1=-1 - \sin(p_1-p_2)\\ & \dot{p}_2=1\\ \end{aligned} $$ I need to find a first integral that is independent of the Hamiltonian. Incidentally, the Hamiltonian is $H=q_1 \sin(p_1-p_2) + q_1 - q_2$.
Thank you, in advance, for your response.