Zoll and Tannery Metrics from a Superintegrable Geodesic Flow

Matveev and Shevchishin have constructed a family of superintegrable models, involving a linear and a cubic integral. Their metrics, when defined on a closed manifold, have the remarkable property of being Zoll metrics. Using the explicit form of the metrics defined on ({mathbb{S}^2}) , which we obtained in a recent paper, we reduce them to a canonical form which displays their Zoll property. For the same class of models several metrics are defined on Tannery’s orbifold and we show, by the same procedure, that in this case we are led either to Zoll or to Tannery surfaces.