Periodic orbits of Euler vector fields on 3-manifolds

In this paper we study the existence of periodic orbits in the flow of non-singular steady Euler fields $X$ on closed 3-manifolds, that is $X$ is a solution of time independent Euler equations. We show, that when $X$ is $C^2$ the flow always posses a periodic orbit unless the manifold is a torus bundle over the circle. This result generalizes preavious result of J. Etnyre and R. Ghrist by weakening the real analytic hypothesis to $C^2$.