  We classify Riemannian surfaces admitting associated families in three
dimensional homogeneous spaces with four-dimensional isometry groups and in a
wide family of (semi-Riemannian) warped products, with an extra natural
condition (namely, rotating structure vector field). We prove that, provided
the surface is not totally umbilical, such families exist in both cases if, and
only if, the ambient manifold is a product and the surface is minimal. In
particular, there exists no associated families of surfaces with rotating
structure vector field in the Heisenberg group.
