We derive necessary and sufficient conditions to test whether a given family of planar curves { f(x,y) = q can be created as orbits traced by a unit mass material point in the presence of an integrable potential V(x,y) with a second integral quadratic in the momenta. Our results are based on two partial differential equations known from the theory of the inverse problem and from Darboux's integrability criterion. We describe an algorithmic procedure for the search of the potential and the corresponding second integral (if it exists) starting from the given family. Several examples illustrate the method.