Integral points of bounded degree on affine curves

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field. Generalizing Picard's theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree d (or less) analytic cover of $\mathbb{C}$.