  We study the effect of symmetry breaking perturbations in the one-dimensional
SU(4) spin-orbital model. We allow the exchange in spin ($J_1$) and orbital
($J_2$) channel to be different and thus reduce the symmetry to SU(2) $\otimes$
SU(2). A magnetic field $h$ along the $S^z$ direction is also applied. Using
the formalism developped by Azaria et al we extend their analysis of the
isotropic $J_1=J_2$, h=0 case and obtain the low-energy effective theory near
the SU(4) point in the asymmetric case. An accurate analysis of the
renormalization group flow is presented with a particular emphasis on the
effect of the anisotropy. In zero magnetic field, we retrieve the same
qualitative low-energy physics than in the isotropic case. In particular, the
massless behavior found on the line $J_1=J_2>K/4$ extends in a large
anisotropic region. We discover though that the anisotropy plays its trick in
allowing non trivial scaling behaviors of the physical quantities. When a
magnetic field is present the effect of the anisotropy is striking. In addition
to the usual commensurate-incommensurate phase transition that occurs in the
spin sector of the theory, we find that the field may induce a second
transition of the KT type in the remaining degrees of freedom to which it does
not couple directly. In this sector, we find that the effective theory is that
of an SO(4) Gross-Neveu model with an h-dependent coupling that may change its
sign as h varies.
