What is the equation of the curve

$$
x^{4}-y^{4}=4\left(1-y^{2}\right)
$$

equation?
We transform the equation into a product form through manipulation.

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

A product is zero if one of its factors is zero. The first factor is 0 if

$$
x^{2}+y^{2}=2
$$

This is the equation of a circle centered at the origin with a radius of $\sqrt{2}$. The second factor is 0 if $y^{2}-x^{2}=2$, which can be written as $\frac{y^{2}}{2}-\frac{x^{2}}{2}=1$.

This is the equation of a central hyperbola with equal axes, whose major axis is the $Y$-axis, and the semi-major axis length is $\sqrt{2}$, so its eccentricity is $\sqrt{\sqrt{2^{2}}+\sqrt{2^{2}}}=2$. The circle and the hyperbola touch each other at the points $(0,+\sqrt{2})$ and $(0,-\sqrt{2})$.