I am trying to find a canonical transformation for this Hamiltonian: \begin{equation} H(q,p) = p + e^p\cos(q) \end{equation} and I'm trying to find a conjugate momentum $P$ to the variable $Q$ defined by \begin{equation} Q(q,p) = e^p\cos(q) \end{equation} I cannot solve the partial differential equations arising from the Poisson brackets, i.e. \begin{equation} \{Q(q,p), P(q,p)\} = 1 \end{equation}
...anyone who can give me a hint on this?