I'm looking for any (non-trivial*) time-independent Hamiltonian expressed in the Pauli basis (with analytically known real coefficients), which unitary time evolves some analytic initial state to some analytic future state. That is, a system for which I can write down the time-evolved state without performing numerics, such as diagonalisation or matrix exponentiation.
*By non-trivial, I mean that the Hamiltonian is not identity, is not all pauli Z (and hence diagonal), and that it consists of multiple terms (since one-term matrices exponentiate trivially). For example, $H = 1 X Y + 2 Z X$ is sufficiently non-trivial. I'm also obviously looking for evolution $t \ne 0$. Furthermore, for the chosen initial state, time evolution shouldn't leave the state unchanged for all time (e.g. like a symmetric Hamiltonian on a uniform state).
This strange request comes from writing a unit test for software which numerically computes time-evolved states of Pauli Hamiltonians. I wish to write a unit test which performs no numerics of its own, and tests whether the software approximates the analytic solution. It's important my "reference" is analytic (and hence precise), since I'll be comparing it to several numerical techniques and must rank their accuracies.
If I knew the eigenspectrum of some Pauli Hamiltonian analytically, then I could evolve each eigenvector separately (as a simple oscillation in phase) and superpose them. Alas, I can't think of any system for which the Pauli coefficients and eigenvectors are simultaneously known.
If I knew (analytically) the period of unitary time evolution of some Hamiltonian (e.g. the product of the eigenvalues, or through other means), then evolving to that period would produce my initial state. But the canonical physical systems with known periods tend to prescribe trivial Pauli Hamiltonians (e.g. the QHO prescribes only Pauli Z).