By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=x'^2+y'^2$.

Question. Suppose the Taylor series of $f$ starts with $(x^2+y^2)^n$. For which $n>1$ one can find coordinates $(x',y')$ so that $f(x',y')=(x'^2+y'^2)^n$?

By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=x'^2+y'^2$.

Question. Suppose now we consider functions $f$ with Taylor series starting with $(x^2+y^2)^n$. For which $n>1$ one can always find coordinates $(x',y')$ so that $f(x',y')=(x'^2+y'^2)^n$?