  In this paper we study surfaces foliated by a uniparametric family of circles
in the homogeneous space Sol$_3$. We prove that there do not exist such
surfaces with zero mean curvature or with zero Gaussian curvature. We extend
this study considering surfaces foliated by geodesics, equidistant lines or
horocycles in totally geodesic planes and we classify all such surfaces under
the assumption of minimality or flatness.
