Evolution equations of the second and third order with Lie-B\"{a}cklund symmetries

TL;DR

This paper discusses the classification of second and third order evolution equations that have Lie-Bäcklund symmetries, based on results from a book's addendum that was previously unavailable in English.

Contribution

It provides the English translation of the addendum containing new classification results for evolution equations with Lie-Bäcklund symmetries.

Findings

Classification of second and third order evolution equations with Lie-Bäcklund symmetries

Summary of new results obtained by 1982

Extension of existing analysis to include recent classifications

Abstract

Chapter 4 of my book [1] is dedicated to Lie-B\"acklund group analysis of various types of differential equations. The Russian edition of this book had an Addendum (pages 262-266) containing a summary of new results (obtained by the end of 1982) on classification of evolution equations of the second and third order possessing Lie-B\"acklund symmetries. The \textit{Addendum} was not included in the English translation of the book by technical reasons. I present here the missing translation. In order to make the text self-contained, I have added in the translation the equations from the main body of the book used in the Addendum.