  The Kardar-Parisi-Zhang (KPZ) equation defines the main universality class
for nonlinear growth and roughening of surfaces. But under certain conditions,
a conserved KPZ equation (cKPZ) is thought to set the universality class
instead. This has non-mean-field behavior only in spatial dimension $d<2$. We
point out here that cKPZ is incomplete: it omits a symmetry-allowed nonlinear
gradient term of the same order as the one retained. Adding this term, we find
a parameter regime where the $1$-loop renormalization group flow diverges. This
suggests a phase transition to a new growth phase, possibly ruled by a strong
coupling fixed point and thus described by a new universality class, for any
$d>1$. In this phase, numerical integration of the model in $d=2$ gives clear
evidence of non mean-field behavior.
