Zoll magnetic systems on the two-torus: a Nash-Moser construction (Q6608723)

\textit{O. Zoll} [Math. Ann. 57, 108--133 (1903; JFM 34.0657.05)] discovered non-trivial examples of surfaces homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length, nowadays named after him. The present paper is a part of ongoing research in this direction and its generalizations.\N\NLet \(\Sigma\) be a closed oriented surface equipped with a Riemannian metric \(g\). A magnetic system on \(\Sigma\) is a pair \((g,f)\), where \(f: \Sigma \to \mathbb{R}\) is a smooth function. A magnetic geodesic is a unit-speed curve \(\gamma: \mathbb{R} \to \Sigma\) satisfying the equation \(\nabla_{\dot \gamma} \dot \gamma = (f \circ \gamma) \cdot {\dot \gamma}^{\perp}\), where \(\nabla\) is the Levi-Civita connection w.r.t. \(g\) and \({\dot \gamma}^{\perp}\) is the unit tangent vector such that the angle between \({\dot \gamma}\) and \({\dot \gamma}^{\perp}\) is \(\tfrac{\pi}{2}\). A magnetic system \((g,f)\) is called Zoll if, up to a smooth time reparametrization, all of its magnetic geodesics are periodic with the same period. In contrast to Zoll metrics, no examples of Zoll magnetic systems have been known so far besides the trivial ones, that is, those with constant Gaussian curvature and a constant function \(f\), or integrable systems on a flat two-torus. The goal of the present paper is to construct an abundance of smooth integrable Zoll magnetic systems on the two-torus, where the metric \(g\) is not flat and the magnetic field \(f\) is not constant.