  Koenigs constructed a family of two dimensional superintegrable (SI) models
with one linear and two quadratic integrals in the momenta, shortly (1,2). More
recently Matveev and Shevchishin have shown that this construction does
generalize to models with one linear and two cubic integrals i.e. $(1,3)$, up
to the solution of a non-linear ordinary differential equation. Our explicit
solution of this equation allowed for the construction of these SI systems and
led to the proof that the systems globally defined on S^2 are Zoll. We will
generalize these results to the case (1,n) for any n \geq 2. Our approach is
again constructive and shows the existence, when n is odd, of metrics globally
defined on S^2 which are indeed Zoll (under appropriate restrictions on the
parameters), while if n is even the metrics we found are never globally defined
on S^2, as it is already the case for the (1,2) models constructed by Koenigs.
