This talk will present several new results about toric degenerations and Hamiltonian symmetries.

First, we will see how gradient-Hamiltonian vector fields can be integrated using Hamiltonian symmetries. This is completely elementary, but very useful when the fibers of a degeneration map are not compact.

Second, we will integrate the gradient-Hamiltonian flows of toric degenerations of the base affine space of an arbitrary complex reductive group. The degenerations we are interested in all have a natural global Hamiltonian torus symmetry. We are able to use this torus symmetry to integrate the gradient-Hamiltonian flows, even though the fibers of the degenerations are all singular and non-compact.

Finally, our results about base affine space will be bootstrapped to new results about arbitrary Hamiltonian spaces for arbitrary compact connected Lie groups. We produce infinitely many examples of integrable torus actions that have never been seen before.

This talk is based on joint work with Benjamin Hoffman.