KAM families of quasi-periodic solutions for spherical water droplets

Let

$$denote the Diophantine equation associated with the spherical water waves equation$$

. KAM conjecture. If the Diophantine equation has only finitely many solutions, then a KAM type result holds for

$$: there is a family of infinitely many quasi-periodic solutions of$$

, depending on a parameter which takes value in a Cantor-type set. This is an ambitious conjecture about the global dynamics of spherical water droplets, motivated by observations in hydrodynamical experiments under zero gravity and by related KAM results. It remains open, conditional on the stated finiteness property of the Diophantine equation.