Infinite-family conjecture for doubly exotic superintegrable systems

Let $N$ be the order of a superintegrable system, let $L_z$ denote the angular momentum, and let $p_1,p_2$ be the Cartesian momenta. Let $V_1$ and $V_2$ be functions of the Cartesian coordinates $x$ and $y$, respectively, and write $V=V_1(x)+V_2(y)$. For a function $\mathcal{G}=\mathcal{G}(u;N)$, let $F_1,F_2,F_3$ be polynomials in $u$ of degree at most $N-1$, and let $\sigma,b$ be real parameters. Infinite-family conjecture. There exists an infinite family of $N$th-order superintegrable systems with $N\geq 5$ and an integral

$${\mathcal Y}_{N}^{(\rm Doubly\ exotic)}=L_z^{(N-4)}p_1^2p_2^2+{\rm(lower\ order\ terms)},$$

satisfying

$$\big\{{\mathcal Y}_{N}^{(\rm Doubly\ exotic)},{\mathcal H}\big\}_{\rm PB}=0.$$

Their associated potential has the form

$$V=V_1(x)+V_2(y)=\mathcal{G}'(x;N)+\mathcal{G}'(y;N),$$

where $\mathcal{G}$ obeys a nonlinear first-order ODE of the form

$${\mathcal{G}}'\big[6u^{N-4}{\mathcal{G}}'+4{(N-5)}u^{N-5}{\mathcal{G}}+F_1(u)+\sigma u^{N-2}\big]+{\mathcal{G}}\big[2{(N-5)}u^{N-6}{\mathcal{G}}+F_2(u)-2\sigma u^{N-3}\big]+F_3(u)+bu^N=0.$$

The conjecture extends the explicitly derived cases, while the detailed conditions under which the closed algebra of integrals is polynomial, and its use as a systematic tool for solving the determining equations, remain for future work.