  Biconservative surfaces of Riemannian 3-space forms $N^3(\rho)$, are either
constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces
verifying the relation $3\kappa_1+\kappa_2=0$ between their principal
curvatures $\kappa_1$ and $\kappa_2$. We characterise the profile curves of the
non-CMC biconservative surfaces as the critical curves for a suitable curvature
energy. Moreover, using this characterisation, we prove the existence of a
discrete biparametric family of closed, i.e. compact without boundary, non-CMC
biconservative surfaces in the round 3-sphere, $S^3(\rho)$. However, none of
these closed surfaces is embedded in $S^3(\rho)$.
