Let $\Omega$ be an $n\times n$ Hermitian matrix with eigenvalues $\omega_1,\ldots,\omega_n$. Consider the n-dimensional Hilbert space, $\mathbb{C}^n$, with the dynamics $|\dot{\psi}\rangle+i\Omega|\psi\rangle=0$. This may viewed as a finite-dimensional quantum system, with Hamiltonian $\Omega$.
For any $(m_1,\ldots,m_n)\in\mathbb{Z}^n$, one can derived the following fact:
\begin{equation} \sum_jm_j\omega_j=0\Longrightarrow\frac{d}{dt}\left(\prod_j\langle\omega_j|\psi\rangle^{m_j}\right)=0 \end{equation}
So, $\Pi_j\langle\omega_j|\psi\rangle^{m_j}$ is a conserved quantity. I think there may even be a true statement along the lines of: two trajectories can be distinguished by a conserved quantity if and only if they have different energy probability distributions, or they have different values for the above quantity using some $\bar{m}\in\mathbb{Z}$.
Since these dynamics are like those that appear in quantum mechanics, it made me wonder if this conserved quantity has a name, and if conserved quantities of this form exist for the Schrodinger equation on infinite dimensional Hilbert spaces.