characteristic system for the linear PDE when some partial derivatives does not appear

Question

The first step to solve the linear Partial Differential Equation $$ f_1(x_1,\dots,x_n)\frac{\partial\omega(x_1,\dots,x_n)}{\partial x_1}+\cdots+f_n(x_1,\dots,x_n)\frac{\partial\omega(x_1,\dots,x_n)}{\partial x_n}=0 $$ is to construct the following characteristic system $$ \frac{dx_1}{f_1(x_1,\dots,x_n)}=\frac{dx_2}{f_2(x_1,\dots,x_n)}=\cdots=\frac{dx_n}{f_n(x_1,\dots,x_n)} $$ What happens to the above characteristic system if some of $f_i$s are zero (i.e. not all $\partial\omega/\partial x_i$ for $i=1,\cdots,n$ does appear in the PDE)?