2. Let the sets be
$$
\begin{array}{l}
A=\left\{x \left\lvert\, x+\frac{5-x}{x-2}=2 \sqrt{x+1}\right., x \in \mathbf{R}\right\}, \\
B=\{x \mid x>2, x \in \mathbf{R}\} .
\end{array}
$$

Then the elements of the set $A \cap B$ are
2. $\frac{5+\sqrt{13}}{2}$.

Notice that, when $x>2$,
$$
x+\frac{5-x}{x-2}=x-2+\frac{x+1}{x-2} \geqslant 2 \sqrt{x+1} \text {. }
$$

By the equality, we have $x-2=\frac{x+1}{x-2}$.
Solving this gives $x=\frac{5+\sqrt{13}}{2}$.