Analogs of the Burgers equation of arbitrary order

Nonlinear equations of the form
$$ \frac{\partial u}{\partial t}=\frac{\partial^n{u}}{\partial x^n}+F\biggl(x,u,\dots,\frac{\partial^{n-1}u}{\partial x^{n-1}}\biggr),\quad n\geqslant 2, $$
are constructed; they are associated with linear substitutions of the Cole–Hopf type and have an infinite set of local symmetries. For $n\leqslant 5$, these equations together with equations of Korteweg–de Vries type exhaust the list of equations of the given type possessing an infinite set of local symmetries.