What is the simplest method to solve the following system? Find all the integers$( x,y,z,t)$ satisfying:
$$xz=a+uk$$
$$yt+xt-yz=-b+vk$$
$$xt+yz=c+wk$$
where $\gcd(a,b,c)=\gcd(u,v,w)=1$, and $ k\neq 0$
Above simultaneous equation shown below:

$xz=a+uk$

$yt+xt-yz=-b+vk$

$xt+yz=c+wk$

Above has numerical solution:

$(x,y,z,t)=(3,2,1,3)$

$(u,v,w,k)=(2,3,5,1)$

$(a,b,c)=(1,-10,6)$
Above simultaneous equation shown below:

$xz=a+uk$

$yt+xt-yz=-b+vk$

$xt+yz=c+wk$

Above has numerical solution:

$(x,y,z,t)=(3,2,1,3)$

$(u,v,w,k)=(2,3,5,1)$

$(a,b,c)=(1,-10,6)$