  The radiative transfer equation (RTE) for polarized light accepts a
convenient exponential solution when the absorption matrix commutes with its
integral. We characterize some of the matrix depth variations which are
compatible with the commutation condition. Eventually the vector solution may
be diagonalized and one may obtain four independent scalar solutions with four
optical depths, complex in general. When the commutation condition is not
satisfied, one must resort to a determination of an appropriate evolution
operator, which is shown to be well determined mathematically, but whose
explicit form is, in general, not easy to apply in a numerical code. However,
we propose here an approach to solve a general case not satisfying the
commutation condition.
