Steady states in Hamiltonian PDEs are often constrained minimizers of energy subject to fixed mass and momentum. I will discuss two examples when the minimizers are degenerate so that spectral stability of minimizers does not imply their nonlinear stability due to lack of coercivity of the second variation of energy. For the example related to the conformally invariant cubic wave equation on three-sphere, we prove that integrability of the normal form equations results in nonlinear stability of steady states. For the other example related to the nonlinear Schrodinger equation on a star graph, we prove that the lack of momentum conservation results in the irreversible drift of steady states and their nonlinear instability.