Solve $ x^2+y^2=4, z^2+t^2=9, xt+yz=6 $ in integers 
find answers of this system of equations in integers$$
\left\{ 
\begin{array}{c}
x^2+y^2=4 \\ 
z^2+t^2=9 \\ 
xt+yz=6
\end{array}
\right. 
$$

things I have done: we can observe that $$(x^2+y^2)(z^2+t^2)=(xt+yz)^2+(xz-yt)^2 \rightarrow xz-yt=0$$ 
summing up this with first and second equality $$(x+z)^2+(y-t)^2=13$$
at this stage I used guessing answers. putting like $x= 0,z=3,y=2,t=0$.is there a better way to doing this without guessing and making sure that all answer found?   

Perhaps you are working too hard. The solutions of the first equation are $x=\pm 2$, $y=0$, or the opposite. The solutions of $z^2+t^2=9$ are equally simple. 
Now look at the third equation.  
