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Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $\sigma$ be anoscillating $2$-form on $\mathbb T^2$. We show that for almost every smallpositive number $k$ the magnetic flow of the pair $(g,\sigma)$ has infinitelymany periodic orbits with energy $k$. This result complements the analogousstatement for closed surfaces of genus at least $2$ [Asselle and Benedetti,Calc. Var. Partial Differential Equations, 2015] and at the same time extendsthe main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J.Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.

On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus