  We study rational curves on algebraic varieties, especially on normal affine
varieties endowed with a $\C^*$-action. For varieties with an isolated
singularity, we show that the presence of sufficiently many rational curves
outside the singular point strongly affects the character of the singularity.
This provides an explanation of classical results due to H.A. Schwartz and G.H.
Halphen on polynomial solutions of the generalized Fermat equation.
