2. The number of pairs of positive integer solutions $(x, y)$ for the equation $x^{3}-y^{3}+x^{2} y-x y^{2}=32$ is ( ).
(A) 0
(B) 1
(C) 2
(D) greater than or equal to 3
2. (B).
$$
\begin{array}{l}
\because x^{3}-y^{3}+x^{2} y-x y^{2} \\
\quad=(x-y)(x+y)^{2}=2^{5},
\end{array}
$$

Since $x, y$ are positive integers, we have
$$
\left\{\begin{array}{l}
x+y=2^{n}, \\
x-y=2^{m}, \\
2 n+m=5 .
\end{array} \quad(n, m \text { are non-negative integers) }\right.
$$

Thus,
$$
\left\{\begin{array}{l}
x=2^{n-1}+2^{m-1} \\
y=2^{n-1}-2^{m-1} \\
2 n+m=5
\end{array}\right.
$$

From (3), if $n=1$, then $m=3$, or $n=2, m=1$, thus we have $x=5, y=-3$, or $x=3, y=1$.
$\therefore$ The only positive integer solution is $(x, y)=(3,1)$.