Haantjes algebras, Zernike system and separation of variables

The generalized Zernike family $H_{(N)} = p_1^2 + p_2^2 + \sum_{n=1}^N \gamma_n\,(q_1 p_1 + q_2 p_2)^n$ is a parametric family of two-dimensional superintegrable Hamiltonians, admitting $N$ integrals of motion of degree $N$ in the momenta. A theorem of Nozaleda, Tempesta, and Tondo guarantees that canonical separation coordinates (Darboux--Haantjes coordinates) exist for any such system; the challenge is to construct them explicitly. This paper solves the problem for $N = 2$ -- the classical Zernike system, which is canonically equivalent to the isotropic harmonic oscillator on flat space or on a space of constant curvature -- covering all four known separation types: polar, two Cartesian-type, and elliptic. The key structural fact is that the Haantjes operators associated with all integrals of $H_{(2)}$ have no momentum-dependent off-diagonal block (lift form). We prove that this implies the separation coordinates are reachable by an extended point transformation: the new positions depend only on the old positions, with no momentum entering the coordinate change. In the polar and Cartesian-type cases the new position coordinates involve at most a square root of a single-variable rational function; in the elliptic case they are given by the two roots of a quadratic polynomial in the original coordinates, and the resulting branch structure introduces a fourth regular singular point in the quantum separated ODE, placing it in the Heun class, in agreement with results of Atakishiyev, Pogosyan, Vicent, Wolf, and Yakhno. For $N \geq 3$ we prove an obstruction: no lift-form Haantjes operator can generate an integral independent of the angular momentum. The separation coordinates for higher Zernike Hamiltonians therefore require momentum-dependent canonical transformations, whose explicit construction is the subject of future work.