Is there any non-constant real analytic periodic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\{x\in\mathbb{R}^n\mid\nabla f(x)=0 \}\subset\{x\in\mathbb{R}^n\mid\nabla^2 f(x)=0 \}? $$ The periodicity of $f$ is that $$f(x+m)=f(x),\quad\forall m\in\mathbb{Z}^n.$$
I can construct examples with the given property but the function $f$ is smooth. For example, let $f(x)=(2-x^4)\operatorname{exp}(1/(x^4-1)), x\in[-1,1]$, then copy and translate it to fill the whole $\mathbb{R}$.