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The stationary Schr\"odinger equation can be cast in the form $H \rho = E\rho$, where $H$ is the system's Hamiltonian and $\rho$ is the system's densitymatrix. We explore the merits of this form of the stationary Schr\"odingerequation, which we refer to as~SSE$_\rho$, applied to many-body systems withsymmetries. For a nondegenerate energy level, the solution $\rho$ of theSSE$_\rho$ is merely a projection on the corresponding eigenvector. However, inthe case of degeneracy $\rho$ is non-unique and not necessarily pure. In fact,it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly,$\rho$ can always be chosen to respect all symmetries of the Hamiltonian, evenif each pure eigenstate in the corresponding degenerate multiplet spontaneouslybreaks the symmetries. This and other features of the solutions of theSSE$_\rho$ can prove helpful by easing the notations and providing anunobscured insight into the structure of the eigenstates. We work out theSSE$_\rho$ for a general system of spins $1/2$ with Heisenberg interactions,and address simple systems of spins $1$. Eigenvalue problem for quantumobservables other than Hamiltonian can also be formulated in terms of densitymatrices. As an illustration, we provide an analytical solution to theeigenproblem ${\bf S}^2 \rho=S(S+1) \rho$, where $\bf S$ is the total spin of$N$ spins $1/2$, and $\rho$ is chosen to be invariant under permutations ofspins. This way we find an explicit form of projections to the invariantsubspaces of ${\bf S}^2$.

Merits of using density matrices instead of wave functions in the stationary Schr\"odinger equation for systems with symmetries