Newton non-degenerate foliations on projective toric surfaces

We prove that the isolated invariant branches of a weak toric type generalized curve defined over a projective toric ambient surfaces extend to projective algebraic curves. To do it, we pass through the characterization of the weak toric type foliations in terms of “Newton non-degeneracy” conditions, in the classical sense of Kouchnirenko and Oka. Finally, under the strongest hypothesis of being a toric type foliation, we find that there is a dichotomy: Either it has rational first integral but does not have isolated invariant branches or it has finitely many global invariant curves and all of them are extending isolated invariant branches.