16. The integer $a$ makes the system of equations about $x, y$
$$
\left\{\begin{array}{l}
x-2 y=3 a-b, \\
x y=b^{2}-2 a^{2}+3 b+4
\end{array}\right.
$$

have real solutions for every real number $b$. Find the value of the integer $a$.
16. Substituting $x=3 a-b+2 y$ into $x y=b^{2}-2 a^{2}+$ $3 b+4$, we get
$$
\begin{array}{l}
2 y^{2}+(3 a-b) y-\left(b^{2}-2 a^{2}+3 b+4\right)=0 . \\
\Delta=(3 a-b)^{2}+8\left(b^{2}-2 a^{2}+3 b+4\right) \\
=9 b^{2}-6(a-4) b-7 a^{2}+32 \\
=(3 b-a+4)^{2}-\left(8 a^{2}-8 a-16\right) \geqslant 0 .
\end{array}
$$

Since for every real number $b$ the system of equations in $x$ and $y$ always has real solutions, we have
$$
\left(8 a^{2}-8 a-16\right) \leqslant 0,
$$

which is $8(a-2)(a+1) \leqslant 0$.
Thus, $-1 \leqslant a \leqslant 2$.
Therefore, the integer values of $a$ are $-1,0,1,2$.