2. The number of all integer pairs $(x, y)$ that satisfy the equation $y^{4}+2 x^{4}+1=4 x^{2} y$ is ( ).
(A) 2
(B) 3
(C) 4
$(\Gamma) 5$
2. (A).

From the given, we have
$$
2\left(x^{2}-y\right)^{2}+\left(y^{2}-1\right)^{2}=0 \text {. }
$$

And $2\left(x^{2}-y\right)^{2} \geqslant 0, y^{2}-1 \geqslant 0$,
$$
\therefore x^{2}-y=0, y^{2}-1=0 \text {. }
$$

The integer pairs that satisfy the conditions are $(1,1),(-1,1)$.