2. The complex number $z$ satisfies $z+\frac{1}{z} \in R$ and $|z-2|=$ $\sqrt{2}$. Then the number of such complex numbers is ( ).
(A) 1
(B) 2
(C) 3 $^{-}$
(D) 4
2. (D).

The complex number $z$ that satisfies $z+\frac{1}{z} \in R$ corresponds to points on the $x$-axis (excluding the origin) or on the unit circle.

The points that satisfy $|z-2|=\sqrt{2}$ lie on a circle with center at $(2,0)$ and radius $\sqrt{2}$.

The latter clearly intersects the former at 4 points (intersecting the unit circle at two points, and the $x$-axis at two points). Therefore, the answer is (D).