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.

We solve the second system for $y$:
$$y=\frac{-x^2+\sqrt{x}+1}{\sqrt{x}+\sqrt{1-x^2}}$$
and substitute into the first equation and solve for $x$. It is then seen that $(0,1)$ is the only real-valued solution. Computer analysis finds complex solutions where $x$ is the root of a certain $16$ degree polynomial.
Edit: the precise polynomial is $$ x^{16}+12 x^{15}+30 x^{14}-96 x^{13}-79 x^{12}+360 x^{11}+70 x^{10}-804 x^9-92 x^8+972 x^7+230 x^6-600 x^5-207 x^4+192 x^3+62 x^2-36 x+1 =0$$
