2. The system of equations
$$
\left\{\begin{array}{l}
\frac{9}{2(x+y)}=\frac{1}{x}+\frac{1}{y}, \\
\sqrt{x^{2}-2}=\sqrt{3-y^{2}}
\end{array}\right.
$$

has the real solutions
2. $(2,1),(-2,-1)$.
$$
\begin{array}{l}
\text { Given } \frac{9}{2(x+y)}=\frac{1}{x}+\frac{1}{y} \\
\Rightarrow 2(x+y)^{2}=9 x y \\
\Rightarrow x=2 y \text { or } y=2 x . \\
\text { Also } \sqrt{x^{2}-2}=\sqrt{3-y^{2}} \Rightarrow x^{2}+y^{2}=5 .
\end{array}
$$
$$
\text { It is easy to see that } y= \pm 1 \text { or } x= \pm 1 \text { (discard). }
$$