  We study the measure of unique information $UI(T:X\setminus Y)$ defined by
Bertschinger et al. (2014) within the framework of information decompositions.
We study uniqueness and support of the solutions to the optimization problem
underlying the definition of $UI$. We identify sufficient conditions for
non-uniqueness of solutions with full support in terms of conditional
independence constraints and in terms of the cardinalities of $T$, $X$ and $Y$.
Our results are based on a reformulation of the first order conditions on the
objective function as rank constraints on a matrix of conditional
probabilities. These results help to speed up the computation of
$UI(T:X\setminus Y)$, most notably when $T$ is binary. In the case that all
variables are binary, we obtain a complete picture of where the optimizing
probability distributions lie.
