  A celebrated theorem of M. Heins says that up to post-composition with a
M\"obius transformation, a finite Blaschke product is uniquely determined by
its critical points. K. Dyakonov suggested that it may interesting to extend
this result to infinite degree, however, one needs to be careful since inner
functions may have identical critical sets.
  In this work, we try parametrizing inner functions by 1-generated invariant
subspaces of the weighted Bergman space $A^2_1$. Our technique is based on the
Liouville correspondence which provides a bridge between complex analysis and
non-linear elliptic PDE.
