I recently came across a system of PDEs $\frac{\partial S}{\partial z}= f_1(x,y,z,w,t)$, $\frac{\partial S}{\partial w}= f_2(x,y,z,w,t)$, $\frac{\partial S}{\partial t}= f_3(x,y,z,w,t)$, $S(x,y,1,1,1)=f_4(x,y)$, where $S$ is an unknown function of five variables $x,y,z,w,t$ and $f_i$ are known. The question is how to obtain a general solution for $S$?

What you have is a family of PDEs labelled by $x,y$ and for fixed $x,y$ you have an equation $$dS = f_1 dz + f_2 dw + f_3 dt$$ for a function $S:\mathbb{R}^3 \to \mathbb{R}$ of the three variables $z,w,t$. The first thing to check is that the equation is integrable: namely, that the 2form $d(dS) = 0$. If that is the case, then $S$ exists up to a locally constant function, which you fix from the fourth condition. To find the explicit solution, choose any path $\gamma$ from $(1,1,1)$ to $(z,w,t)$ and integrate the righthand side of the equation: $$S(x,y,z,w,t) = S(x,y,1,1,1) + \int_\gamma (f_1 dz + f_2 dw + f_3 dt).$$ 


By using method of characteristics. See the book of Zachmanoglous Thoe " Introduction to Partial differential equations with applications" or Evans "Partial differential equations" GSM 19 AMS 

