  Recently, the nodal domain counts of planar, integrable billiards with
Dirichlet boundary conditions were shown to satisfy certain difference
equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these
equations give the number of domains explicitly. For complete generality, we
demonstrate this novel formulation for three additional separable systems and
thus extend the statement to all integrable billiards.
