  We present a family of superintegrable (SI) sytems living on a riemannian
surface of revolution and which exhibits one linear integral and two integrals
of any integer degree larger or equal to 2 in the momenta. When this degree is
2 one recovers a metric due to Koenigs. The local structure of these systems is
under control of a linear ordinary differential equation of order n which is
homogeneous for even integrals and weakly inhomogeneous for odd integrals. The
form of the integrals is explicitly given in the so-called simple case (see
definition 2). Some globally defined examples are worked out which live either
in H2 or in R2.
