Integrability classification conjecture for the minimally coupled scalar-field system

Let

$$H = \frac12 \left(-p_1^2 + \frac{1}{q_1^2}p_2^2 \right) - k q_1^2 + \Lambda q_1^4 + m^2q_2^2 q_1^4,$$

be the Hamiltonian of the minimally coupled scalar-field system. Assume $\Lambda\ne0$, let $n$ be an integer satisfying

$$9-\frac{4m^2}{\Lambda}=(2n+1)^2,$$

and consider a generic energy level $E\ne0$. Integrability classification conjecture. If the system is integrable, then one of the following holds: $n=1$ or $n=-2$, with $m=0$ in both cases; $k=0$ and $9-4m^2/\Lambda=(2l)^2$ for an odd integer $l$; or $k=0$ and

$$n+\frac12\in \frac13\mathbb Z\cup\frac14\mathbb Z\cup\frac15\mathbb Z\setminus\mathbb Z.$$

This statement appears in the conclusions as a classification of the remaining potentially integrable parameter values on generic nonzero energy levels, complementing the necessary conditions proved earlier. Its resolution is not supplied in the paper.