We get negative Gauss curvature K surfaces of quasi polar symmetry surface form generated from:

$$ Re (x+ i y)^n = a^n $$ (n integer) with n humps above plane $ z =0$.

($ n =2,3,4 $ hyperbolic paraboloids, monkey saddles, four humped frill/ pleated shells respectively).

"Quasi" due to $n>2$ introducing circumferential pleats or frills.

What polynomial or differential relation function yields constant negative surfaces of $K = -1/a^2 $?

How to set up its differential equation?

Instead of $ z^n $ if a simpler $ \cos z $ is taken, no clue or handle how to modify the function in order to get a constant negative K.