5. Given the system of equations in terms of $x$ and $y$
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=a-1 \\
x y=a-2
\end{array}\right.
$$

has two real solutions. Then which of the following statements is correct? ( )
(A) $a$ can only be 2
(B) $a$ can only be $\frac{5}{3}$
(C) $a$ can only be 3
(D) None of the above
5. D.

Team Fire Qian Group is symmetric in $x, y$ of the second degree, so if the system of equations has one solution $(s, t)$, there must be three other solutions $(-s,-t), (t, s)$, and $(-t,-s)$.

For the original system of equations to have two solutions, it must be that $s=t$ or $s=-t$, i.e., $x=y$ or $x=-y$.
When $x=y$, we get $a=3$, at which point the two solutions are
$$
\left\{\begin{array} { l } 
{ x = 1 } \\
{ y = 1 }
\end{array} \left\{\begin{array}{l}
x=-1 \\
y=-1
\end{array}\right.\right.
$$

When $x=-y$, $a=\frac{5}{3}$, at which point the two solutions are

Therefore, $a$ can only be 3 or $\frac{5}{3}$.