Following work of Harada--Kaveh, gradient-Hamiltonian flows have been used to construct integrable systems on many smooth projective varieties. However, from the perspective of a symplectic geometer it is often unnatural to work with only projective varieties. For instance, all coajoint orbits of the unitary group have a densely defined completely integrable torus action, whereas few of these orbits carry a symplectic form coming from an embedding into projective space.

I will give a new approach to using gradient-Hamiltonian flows to build torus actions on symplectic manifolds, which does not require an embedding into projective space. As a consequence, we obtain a generalization of the Gelfand--Zeitlin system for all coadjoint orbits of any compact Lie group. This is joint work with Jeremy Lane.