  Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $\sigma$ be an
oscillating $2$-form on $\mathbb T^2$. We show that for almost every small
positive number $k$ the magnetic flow of the pair $(g,\sigma)$ 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,
Calc. Var. Partial Differential Equations, 2015] and at the same time extends
the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J.
Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case.
