  For a given simply connected Riemannian surface Sigma, we relate the problem
of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a
system of two partial differential equations on Sigma. We prove that a constant
intrinsic curvature minimal surface in S^2 x R or H^2 x R is either totally
geodesic or part of an associate surface of a certain limit of catenoids in H^2
x R. We also prove that if a non constant curvature Riemannian surface admits a
continuous one-parameter family of minimal isometric immersions into S^2 x R or
H^2 x R, then all these immersions are associate.
