I have a Hamiltonian
$$ H = \sum_{1\leq i < j \leq n} p_i p_j \frac{\sinh({\lambda} (q_i - q_j))}{\lambda^2}, $$ where ($\textbf{q},\textbf{p}) \in \mathbb{R}^{2n}$, and $\lambda$ is a free parameter non equal to zero. For the reduction method I tried to guess the "first" first integral. Something like this $$ I_{kl} = p_k p_l \frac{\sinh({\lambda} (q_k - q_l))}{\lambda^2} $$ does not commute(in terms of poisson-braket) with Hamiltonian. Don't know what to do with this one. Any help would be appreciated.