How to solve this nonstandard system of equations? How to solve this system of equations
$$\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1+(1-y)\sqrt{x}.
\end{cases}$$
I see $(0,1)$ is a root.

(More a comment.) If we allow the non-principal square root,
$$\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1\color{red}{\pm} (1-y)\sqrt{x}
\end{cases}$$
the $+$ case has one real solution, but the $-$ case has three real solutions: $(x,y)=(0,1)$ and two which surprisingly are roots of $11$-deg equations. Using,
$$x=+\sqrt{\frac{1-y^2}{2}}$$
and with $y$ as two appropriate real roots of,
$$\small49 - 301 y + 327 y^2 - 51 y^3 - 1038 y^4 + 1334 y^5 - 1066 y^6 + 850 y^7 - 259 y^8 + 279 y^9 + 3 y^{10} + y^{11}=0$$
namely $y_1 \approx -0.619107$, and $y_2 \approx 0.939251$.
