On the Classification of Scalar Non-Polynomial Evolution Equations: Quasilinearity

We prove that, for $m\ge 7$, scalar evolution equations of the form $u_t=F(x,t,u,\mathrm{.}\mathrm{.}\mathrm{.},u_m)$ which admit a nontrivial conserved density of order $m+1$ are linear in $u_m$. The existence of such conserved densities is a necesary condition for integrability in the sense of admitting a formal symmetry, hence integrable scalar evolution equations of order $m\ge 7$ are quasilinear.