This paper gives a new, explicit description of all rotationally symmetric Zoll metrics on the 2‑sphere that have an extra conserved quantity which is cubic in the momenta (that is, a polynomial of degree three in velocities). A Zoll metric is a Riemannian metric on the sphere for which every geodesic is closed and all closed geodesics have the same length. The authors show that every such rotationally symmetric metric with a non‑trivial cubic integral is isometric to one member of a two‑parameter family they write down in closed form.

The starting point is Funk’s form for a metric of revolution on the sphere. In these coordinates the metric is determined by a single function h(x) with a list of regularity and symmetry conditions: h is smooth, odd in x, and satisfies specific endpoint and positivity constraints that make the metric Zoll. Any such metric automatically has two basic conserved quantities for geodesic motion: the energy (quadratic in the momenta) and the angular momentum (linear in the momenta). The question the paper addresses is which choices of h(x) admit a third, independent conserved quantity that is cubic in the momenta.

The authors introduce an algebraic route that avoids solving difficult differential equations. They use Funk’s metric together with an algebraic relation between integrals. By building two complex eigenfunctions of a certain linear operator (denoted F_+ and F_-) and then forming the product P = F_+ F_-, they obtain a function that is independent of the angular coordinate and commutes with the other conserved quantities. They prove that P must be a “Casimir’’—a polynomial function of the energy H and angular momentum L of the specific form P = c0 H^3 + c1 H^2 L^2 + c2 H L^4 + c3 L^6 with constants c0,…,c3. This algebraic relation reduces the problem to finite algebraic equations for the unknown function h(x), which the authors solve explicitly.