1. Let $\log _{2} x=m \in Z, m>0, \log _{6} y=n \in Z, n>0$.

Then $x=2^{m}, y=6^{n}$. As a result, we have

$$
\text { GCD }(x, y)=\text { GCD }\left(2^{m}, 6^{n}\right)=\text { GCD }\left(2^{m}, 2^{n} \cdot 3^{n}\right)=8=2^{3} .
$$

Case 1. $m \geq n$. Then $n=3, \quad y=6^{3}=216$,

GCD $\left(\log _{2} x, 3\right)=3$. From this, we find $\log x=B k=$ or $x=8^{k}, k \in Z$.

Case 2. $m0$ or $y=216^{s}, s \in Z, s>0$.
Answer: 1) $x=8^{k}, k=1,2, \ldots ; y=216$;
2) $x=8, y=216^{s}, s=1,2, \ldots$