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We construct an infinite-dimensional family of smooth integrable magneticsystems on the two-torus which are Zoll, meaning that all the unit-speedmagnetic geodesics are periodic. The metric and the magnetic field of suchsystems are arbitrarily close to the flat metric and to a given constantmagnetic field. This extends to the magnetic setting a famous result byGuillemin on the two-sphere. We characterize Zoll magnetic systems as zeros ofa suitable action functional $S$, and then look for its zeros by means of aNash-Moser implicit function theorem. This requires showing theright-invertibility of the linearized operator $\mathrm{d} S$ in a neighborhoodof the flat metric and constant magnetic field, and establishing tame estimatesfor the right inverse. As key step we prove the invertibility of the normaloperator $\mathrm{d} S\circ \mathrm{d} S^*$ which, unlike in Guillemin's case,is pseudo-differential only at the highest order. We overcome this difficultynoting that, by the asymptotic properties of Bessel functions, the lower orderexpansion of $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integraloperators. We then use a resolvent identity decomposition which reduces theproblem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ restrictedto the subspace of functions corresponding to high Fourier modes. The inversionof such a restricted operator is finally achieved by making the crucialobservation that lower order Fourier integral operators satisfy asymmetric tameestimates.

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