The distance among two counter-rotating vortex filaments satisfies a beam-type of equation according to the model derived by Damodaran, Klein and Majda. This equation has an explicit solution where two straight filaments travel with constant speed at a constant distance. The boundary condition of the filaments is $2\pi$-periodic. Using the distance of the filaments as bifurcating parameter, an infinite number of branches of periodic standing waves bifurcate from this initial configuration with constant rational frequency along each branch.