Integration of the radiative transfer equation for polarized light: the exponential solution

The radiative transfer equation (RTE) for polarized light accepts a conveni ent exponential solution when the absorption matrix commutes with its integ ral. We characterize some of the matrix depth variations which are compatib le with the commutation condition. Eventually the vector solution may be di agonalized and one may obtain four independent scalar solutions with four o ptical depths, complex in general. When the commutation condition is not sa tisfied, one must resort to a determination of an appropriate evolution ope rator, which is shown to be well determined mathematically, but whose expli cit 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 commut ation condition.