If $a,b,c$are all different and the equations $ax+a^{2}y+(a^3+1)z=0,$$bx+b^{2}y+(b^3+1)z=0,$$cx+c^{2}y+(c^3+1)z=0$

 have a non-zero solution, then

$Given equations have non-zero solution\Rightarrow \Delta =0$

$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}a& a^2& a^3+1\\ b& b^2& b^3+1\\ c& c^2& c^3+1\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}a& a^2& a^3\\ b& b^2& b^3\\ c& c^2& c^3\end{array}\right|+\left|\begin{array}{ccc}a& a^2& 1\\ b& b^2& 1\\ c& c^2& 1\end{array}\right|=0 \\ \Rightarrow abc\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|+\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=0 \\ \Rightarrow \left(abc+1\right)\left(a-b\right)\left(b-c\right)\left(c-a\right)=0 \\ \Rightarrow abc+1=0 \left(\because a, b, c are different\right) \end{gathered}$$