  We provide and study complete sets of one-loop renormalization group
equations of several Finkel'stein non-linear $\sigma$-models, the effective
field theories describing the diffusive quantum fluctuations in correlated
disordered systems. We consider different cases according to the presence of
certain symmetries induced by the original random Hamiltonians, and we show
that, for interacting systems, the Cartan's classification of symmetry classes
is not enough to uniquely determine their scaling behaviors.
