  In this paper, we construct and classify minimal surfaces foliated by
horizontal constant curvature curves in product manifolds $M \times \R$, where
$M$ is the hyperbolic plane, the Euclidean plane or the two dimensional sphere.
The main tool is the existence of a Jacobi field which characterize the
property to be foliated in circles and geodesics in these product manifolds. It
is related to harmonic maps.
