Let (T2, g) be a Riemannian two-torus and let σ be an oscillating 2-form on T2. We show that for almost every small positive number k the magnetic flow of the pair (g, σ) has infinitely many periodic orbits with energy k. This result complements the analogous statement for closed surfaces of genus at least 2 (Asselle and Benedetti in Calc Var Partial Differ Equ 54(2):1525–1545. doi:10.1007/s00526-015-0834-1, 2015) and at the same time extends the main theorem of Abbondandolo et al. (J Eur Math Soc, arXiv:1404.7641, to appear) to the non-exact oscillating case.
