In this paper, we study integral points and generators on quartic curves of the forms u2 ± v4=m for a nonzero integer m. The main results assert that certain integral points on the curves can be extended to bases for the Mordell-Weil groups of the elliptic curves attached to the quartic curves in the cases where the Mordell-Weil ranks are at most two. As corollaries, we explicitly describe the integral points on the quartic curves in each case where the ranks are one and two.