1. $5|\cos 2 x|$ from natural values can only take the values $1,2,3,4,5$. Since

$\text{GCD}(5|\cos 2 x|, a|\sin 2 x|)=3$

then $5|\cos 2 x|=3$, i.e.

$|\cos 2 x|=3 / 5 \Rightarrow|\sin 2 x|=4 / 5$ and $x= \pm \frac{1}{2} \arccos \frac{3}{5}+\frac{\pi n}{2}, n \in Z$.

For the parameter $a$ we get the equation

$$
\text{GCD}(3,4 a / 5)=3,
$$

which is equivalent to $a$ being a multiple of 15.
Answer: $\left\{\begin{array}{l}a=15 k, k \in N ; \\ x= \pm \arccos \frac{3}{5}+\frac{\pi n}{2}, n \in Z .\end{array}\right.$