  We find a connection between the existence of a factorization of a quantum
channel and the existence of low-rank solutions to certain linear matrix
equations. Using this, we show that if a quantum channel is factorized by a
direct integral of factors, it must lie in the convex hull of quantum channels
which are factorized respectively by the factors in the direct integral. We use
this to characterize some non-trivial extreme points in the set of factorizable
quantum channels and give an example.
