We construct nontrivial deformations of the standard map that preserve the symplectic actions of infinitely many periodic orbits accumulating on an invariant curve with a Liouvillean rotation number. This result can be viewed as an analogue of a spectral nonrigidity phenomenon for convex billiards. After a brief review of the background and related results, I will discuss key components of the proof, including a resonant normal-form construction that captures the dependence of periodic orbits on the resonant Fourier coefficients of the dynamics.