The action of a quantum channel $\mathcal{E}$ on state given by a density operator $\rho$ is described by a completely positive trace preserving (CPTP) map: $$ \rho \rightarrow \sum_k R_k \rho R_k^\dagger = \mathcal{E(\rho)} . $$ The Kraus operators $R_k$ must obey the condition $\sum_kR_k^\dagger R_k = \mathcal{I}$ where $\mathcal{I}$ is the identity operator.
A fixed point $\rho_f$ of a channel is a state which is left unchanged by the action of the channel so that: $$ \mathcal{E}(\rho_f) = \rho_f $$ My question is: Given that we know the Kraus operators $R_k$ for a quantum channel $\mathcal{E}$, how can we find the fixed point(s) of that channel?
Is there a general process for obtaining an expression for $\rho_f$ in terms of $R_k$?
A related, maybe easier, question is whether we can prove that there exists a fixed point of a particular channel. Do all channels have fixed points?