(M) Find the $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f\left(x^{4}+4 y^{4}\right)=f\left(x^{2}\right)^{2}+4 y^{3} f(y)
$$
$y=0$ gives $f\left(x^{4}\right)=f\left(x^{2}\right)^{2}$ in particular we have $x \geqslant 0$ implies $f(x) \geqslant 0$ and $x=0$ gives $f\left(4 y^{4}\right)=f(0)^{2}+4 y^{3} f(y)$ and with $y$ such that $4 y^{3}=1$ we have $f(0)=0$ we therefore have the oddness of $f$. Then the basic equality rewrites for all $x, y \geqslant 0$ we have $f(x+y)=f(x)+f(y)$ and thus $f$ is increasing on $\mathbb{R}^{+}$. And thus $f$ is linear on $\mathbb{R}^{+}$ and therefore on $\mathbb{R}$ by oddness and thus $f=I d$.