If $p\ne a,q\ne b,r\ne c$  and the system of equations
$px+ay+az=0$ ,
$bx+qy+bz=0$ ,
 $cx+cy+rz=0$
has only a trivial solution, then a possible value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$  is  $\lambda$,where  $\lambda \in R$ .Then the maximum possible integral value of $x,$ where $x\in [-1,2]$  which does not satisfy equation $x^3-3x=\lambda$  is ___

As the given system of equations has a non-trivial solution  $\Delta =\left|\begin{array}{ccc}p& a& a\\ b& q& b\\ c& c& r\end{array}\right|=0$
Applying  $C_2\to C_2-C_1$ and $C_3\to C_3-C_2$ , we get $\Delta =\left|\begin{array}{ccc}p& a-p& a-p\\ b& q-b& 0\\ c& 0& r-c\end{array}\right|=0$  expanding along $C_3{C}_3$, we get expanding along $C_3$ , we get  ${(a-p)}_c^b{}_0^{q-b}+(r-c)\left|\begin{array}{cc}p& a-p\\ b& q-b\end{array}\right|=0$
 $\Rightarrow (a-p)(-c)(q-b)+(r-c)\{p(q-b)-b(a-p)\}=0$
 $\Rightarrow (p-a)(q-b)c+p(r-c)(q-b)+b(r-c)(p-a)=0$
Diving by v we get  $\frac{c}{r-c}+\frac{p}{p-a}+\frac{b}{q-b}=0$
 $\Rightarrow \frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}-\frac{q-b}{q-b}+\frac{r-c}{r-c}-2$