Suppose we allow two integrable systems with Hamiltonians $H_1$ and $H_2$ to interact. Then their combined dynamics can be described by a joint Hamiltonian, $$H = H_1(\mathbf{q}_1,\mathbf{p}_1) + H_2(\mathbf{q}_2,\mathbf{p}_2) + h(\mathbf{q}_1,\mathbf{q}_2),$$ where $h$ is an interaction term. We have assumed for simplicity that the coupling function depends only on the generalised coordinates $\mathbf{q}_1$ and $\mathbf{q}_2$.
Is it true that $H$ is integrable if $h$ is a linear combination of $\mathbf{q}_1$ and $\mathbf{q}_2$? If so, how can this be shown?
I can think of two possible approaches: (1) showing that the Hamilton-Jacobi equation is separable for $H$ if it is separable for $H_1$ and $H_2$, or (2) constructing $2n$ first integrals of $H$ from each of the $n$ first integrals of $H_1$ and $H_2$. I'm very new to all this, so I don't know which approach is best (if either) or exactly how to go about them.
