  We consider invariant matrix models with log-normal (asymptotic) weight. It
is known that their eigenvalue distribution is intermediate between
Wigner-Dyson and Poissonian, which candidates these models for describing a
system intermediate between the extended and localized phase. We show that they
have a much richer energy landscape than expected, with their partition
functions decomposable in a large number of equilibrium configurations, growing
exponentially with the matrix rank. Within each of these saddle points,
eigenvalues are uncorrelated and confined by a different potential felt by each
eigenvalue. The equilibrium positions induced by the potentials differ in
different saddles. Instantons connecting the different equilibrium
configurations are responsible for the correlations between the eigenvalues. We
argue that these instantons can be linked to the SU(2) components in which the
rotational symmetry can be decomposed, paving the way to understand the
conjectured critical breaking of U(N) symmetry in these invariant models.
