stationary points revisited

I have this function and I am looking for the stationary points:

$\displaystyle

f(x,y) = x^4 + 2x^2y^2 + y^4 + 2x^2 - 2y^2 + 6

$

I have the two partial derivatives:

$\displaystyle

4x(x^2 + y^2 + 1) = 0, (E.1)

$

$\displaystyle

4y(y^2 + x^2 - 1) = 0, (E.2)

$

I have already found the stationary points: $\displaystyle (0,0), (0,1)$ and $\displaystyle (0,-1)$

For $\displaystyle (E.2)$ can we set:

$\displaystyle

y^2 + x^2 = 1

$

and thus have:

$\displaystyle

( \sqrt{1/2}, \sqrt{1/2} )

$

or is this useless because it doesn't satisfy $\displaystyle (E.1)$ ?