Limit cycles, invariant meridians and parallels for polynomial vector fields on the torus

We study the polynomial vector fields of arbitrary degree in R3R3 having the 2-dimensional torusT2={(x,y,z)∈R3:(x2+y2−a2)2+z2=1}with a>1, invariant by their flow. We characterize all the possible configurations of invariant meridians and parallels that these vector fields can exhibit. Furthermore we analyze when these invariant either meridians or parallels can be limit cycles.