Solvability in classical mechanics and algebraic Heun observables

We construct a classical analog of the algebraic Heun operator $W$ associated to any bispectral pair of the operators $X$ and $Y$. We show that the dynamics of the classical variables $X$ or $Y$ is governed by elliptic functions if the classical $W$ is taken as the Hamiltonian. We demonstrate this general property by concrete examples of classical dynamical systems.