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There is a complete description of the direction fields on $\mathbb{T}^2$ with and without periodic orbits in Chapter IV of the book Dynamical Systems on Surfaces by Claude Godbillon. I am in the process of restating the main result for the format of MathOverflow currently. The idea is to reduce a direction field on $\mathbb{T}^2$ with a periodic orbit to an orientable direction field on an annulus.
