Is it decidable when the following system of equation has solutions over the integers?
$$ \begin{matrix} x_1y_2-x_2y_1=a_1,\\ x_1y_3-x_3y_1=a_2,\\ x_2y_3-x_3y_2=a_3. \end{matrix} $$
Here $a_1$, $a_2$ and $a_3$ are integers which $GCD(a_1,a_2,a_3)=1$ and $x_1,x_2,x_3,y_1,y_2,y_3$ are variables.