Rotationally Symmetric Zoll Metrics with a Cubic Integral

Abstract

We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.