A system of two nonlinear algebraic equations Is it possible to determine real solutions $x,y$ for the following system of nonlinear equations in terms of the real constants $a,b,k$?
$$
a=kx(1-y^2)\qquad\mbox{and}\qquad b=ky(1-x^2).
$$
If so, what is (are) the solution(s)?

Square the first equation $a^2 = k^2x^2(1-y^2)^2$. Now $k^2(1-y^2)^2$ minus this equation gives
\begin{eqnarray*}
k^2(1-y^2)^2-a^2 = k^2(1-y^2)^2(1-x^2)
\end{eqnarray*}
Multiply this by $y$ and substitute for $ky(1-x^2)=b$ and we have
\begin{eqnarray*}
\color{red}{y(k^2(1-y^2)^2-a^2) = kb(1-y^2)^2}.
\end{eqnarray*}
This is a quintic (order $5$) in $y$ and will not be easy to solve.
