Impossibility of Stokes decomposition for a class of light beams

Decomposition throughout all space of a typical light beam into the sum of a completely polarized and a completely unpolarized beam (Stokes decomposition) seems to be rather the exception than the rule. However, it is known that such decomposition can often be made across a certain plane, although it may lose validity upon propagation. Then, one may guess that, for any light beam, there exists at least one plane where the Stokes decomposition can be performed. Without adopting any particular model for polarized and unpolarized beams, we present a class of beams for which no such plane can exist.