The 2D Smorodinsky--Winternitz II system and the Laguerre--Heun algebra

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky–Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator $$Y={\mathrm{\partial }}_y^2-{\omega }^2{y}^2+\frac{1\mathrm{/}4-c^2}{y^2}$$ is of Laguerre type, while the parabolic integral $W=L_2$ is its algebraic Heun partner. With $Z=[Y,W]$, the defining relations are $$[Y,Z]=16{\omega }^{2}W-2bY,[W,Z]=6Y^2-4HY+2bW+8{\omega }^2(1-c^2),$$ where $H$ is central. This gives a direct superintegrable realization of the Laguerre–Heun algebra.