3. Express $x$ from the 3rd equation

$$
x=-3 z+5
$$

Substitute it into the 1st equation

$$
\left\{\begin{array}{r}
-3 z+2 y=a-5 \\
2 y+3 z=a+1
\end{array}\right.
$$

Adding the 1st and 2nd equations, we get $4 y=2 a-2$. From this, we find $y=\frac{a-2}{2}$. Then $z=\frac{a+1-2 y}{3}=1$, and $x=2$.

Substitute the solution we found into the equation of the sphere

$$
1+\frac{(a+2)^{2}}{4}+9=19
$$

Rewrite this equation as $(a+2)^{2}=36$. From this, we find $a_{1}=4, a_{2}=-8$. For $a_{1}=4$, we get $x=2, y=1, z=1$, and for $a_{2}=-8 \quad x=2, y=-5, z=1$.
Answer: 1) when $a=4, x=2, y=1, z=1$;

2) when $a=-8 \quad x=2, y=-5, z=1$.