Monkey saddle ($x^3-3x{y}^2$)

This cubic smooth function has one isolated degenerate critical point at the origin, with three ascending and three descending sectors. Adding $-tx$ with $t>0$ replaces it by two nondegenerate saddle points at $(\pm \sqrt{t/3},0)$. The Hessians there have opposite diagonal entries $6x$ and $-6x$. It illustrates why the number of smooth critical points can be smaller than that required for a Morse function.