BALTIC-WAY 2020 Problem 5

Find all real numbers $x, y, z$ so that

$$\begin{aligned} x^{2} y+y^{2} z+z^{2} & =0 \\ z^{3}+z^{2} y+z y^{3}+x^{2} y & =\frac{1}{4}\left(x^{4}+y^{4}\right) \end{aligned}$$