  Polarized NLTE radiative transfer in the presence of scattering in spectral
lines and/or in continua may be cast in a so-called reduced form for six
reduced components of the radiation field. In this formalism the six components
of the reduced source function are angle-independent quantities. It thus
reduces drastically the storage requirement of numerical codes. This approach
encounters a fundamental problem when the medium is illuminated by a polarized
incident radiation, because there is a priori no way of relating the known (and
measurable) Stokes parameters of the incident radiation to boundary conditions
for the reduced equations. The origin of this problem is that there is no
unique way of deriving the radiation reduced components from its Stokes
parameters (only the inverse operation is clearly defined). The method proposed
here aims at enabling to work with arbitrary incident radiation field
(polarized or unpolarized). In previous works an ad-hoc treatment of the
boundary conditions, applying to cases where the incident radiation is
unpolarized, has been used. In this note we show that it is possible to account
for the incident radiation in a rigorous way, without any assumption on its
properties, by expressing the radiation field as the sum of a directly
transmitted radiation and of a so-called diffuse radiation. The diffuse
radiation field obeys a transfer equation with no incident radiation that may
be solved in the reduced form. The first scattering of the incident radiation
introduces primary creation terms in the six components of the reduced source
function. Once the reduced polarized transfer problem is solved for the diffuse
radiation field, its Stokes parameters can be computed. We perform numerical
computations of such cases, showing that the emergent line-polarization may be
significantly affected by the polarization of the incident radiation.
