  Motivated by experiments of scanning tunneling spectroscopy (STS) on
self-assembled networks of iron(II)-phtalocyanine (FePc) molecules deposited on
a clean Au(111) surface [FePc/Au(111)] and its explanation in terms of the
extension of the impurity SU(4) Anderson model to the lattice in the Kondo
regime, we study the competition between the Kondo effect and the
magneto-orbital interactions occurring in FePc/Au(111). We explore the quantum
phases and critical points of the model using a large-$N$ slave-boson method in
the mean-field approximation. The SU(4) symmetry in the impurity appears as a
combination of the usual spin and an orbital pseudospin arising from the
degenerate $3d_{xz}$ and $3d_{yz}$ orbitals in the Fe atom. In the case of the
lattice, our results show that the additional orbital degrees of freedom
crucially modify the low-temperature phase diagram, and induce new types of
orbital interactions among the Fe atoms, which can potentially stabilize exotic
quantum phases with magnetic and orbital order. The dominant instability
corresponds to spin ferromagnetic and orbital antiferromagnetic order.
