$u:\mathbb{R}^4\to \mathbb{R} $ is a given function, where we can note the variables like $u(x,y,z,t)$ and the derivatives as $u_y= \frac{\partial u}{\partial y}$
We are also given $u(x,y,z,0)= f(x,y,z)$ and $u_t(x,y,z,0)= g(x,y,z)$ as two known functions.
We have now $3$ functions $q,r,s$ , also $\mathbb{R}^4\to \mathbb{R}$ , which satisfy the following equations:
$$q_x+r_y+s_z= u_t \\ q_t= u_x \\ r_t= u_y \\ s_t= u_z$$
One can notice that there are infinitely many such triplets of functions that satisfy these equations, but I am only interested in finding one triplet, furthermore I am only interested in determining $q(x,y,z, 0) ,r(x,y,z, 0) $ and $s(x,y,z, 0)$ from that triplet.
I am looking for $q(x,y,z, 0) ,r(x,y,z, 0) $ and $s(x,y,z, 0)$ in terms of $f$ and $g$ .