Given a superoperator $\mathcal{E}$ given by a set of matrices $E_i$ with $\sum_i E_i^* E_i=I$, and a subspace $X$, the question is to check whether there is some density operator $\rho$ such that
(1) $\mathcal{E}(\rho)=\rho$, i.e., $\sum_i E_i\rho E_i^*=\rho$, and
(2) the support of $\rho$ is contained by $X$, where the support of $\rho$ is the subspace spanned by the eigenvectors of $\rho$ which are associated non-zero eigenvalues.
How to solve this question, in particular, is it possible to reduce this problem to a form of semidefinite programs?