I would like to find a reference to the following statement:
Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone of curves of $X$ is generated by $(\mathbb C^*)^k$-invariant curves. In particular, each irreducible proper curve in $X$ is linearly equivalent to a sum of such invariant curves with positive coefficients.
(Note, of course, that for toric manifolds - when the action of $(\mathbb C^*)^k$ has an open orbit - this statement is classical.)
PS. I am aware of a proof of this statement - along the lines of the comment below, but would be especially grateful for a precise reference.