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. ( \cos x \cosh y = $ const)

[![Z= cos_x cosh_y][1]][1]


  [1]: https://i.sstatic.net/XzoUr.png