  We consider the differential equation that Zernike proposed to classify
aberrations of wavefronts in a circular pupil, whose value at the boundary can
be nonzero. On this account the quantum Zernike system, where that differential
equation is seen as a Schr\"odinger equation with a potential, is special in
that it has a potential and boundary condition that are not standard in quantum
mechanics. We project the disk on a half-sphere and there we find that, in
addition to polar coordinates, this system separates in two additional
coordinate systems (non-orthogonal on the pupil disk), which lead to
Schr\"odinger-type equations with P\"oschl-Teller potentials, whose
eigen-solutions involve Legendre, Gegenbauer and Jacobi polynomials. This
provides new expressions for separated polynomial solutions of the original
Zernike system that are real. The operators which provide the separation
constants are found to participate in a superintegrable cubic Higgs algebra.
