I suppose you mean du/ds in the last equation. There is one problem with reparametrizing the characteristic curves: the vector field f is zero at the fixed point, so you are dividing by ||f(x)|| which tends to zero (assuming suitable smoothness of f).

Apart from that you may reparametrize the characteristic curves as you see fit. Notice that this doesn't solve the original 1st order partial differential equation, because to achieve that you still need to compute all solutions to the dynamical system determined by f.

I suppose you mean du/ds in the last equation. There is one problem with reparametrizing the characteristic curves: the vector field f is zero at the fixed point, so you are dividing by ||f(x)|| which tends to zero (assuming suitable smoothness of f). So either you consider only a region that doesn't include the fixed point, or you have to be careful with continuity to ensure that all constructions exist.

Apart from that you may reparametrize the characteristic curves as you see fit. Notice that this doesn't solve the original 1st order partial differential equation, because to achieve that you still need to compute all solutions to the dynamical system determined by f.