The generalized Pillai equation $\pm r{a}^x\pm s{b}^y=c$, II

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $innonnegativeintegers$x, y$andintegers$u, v \in {0,1}$,forgivenintegers$a>1$,$b>1$,$c>0$,$r>0$and$s>0$\mathrm{.}When$(ra,sb)=1$,weshowthat$N \le 3$ except for a finite number of cases all of which satisfy $\max (a,b,r,s,x,y)<2\cdot {10}^{15}$ for each solution; when $(a,b)>1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of infinite families of cases giving N=3 solutions.