I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain.

I also know that a lot of articles related to "classification of differential equations via finding corresponding Lie algebras" are published every year.

  1. What is the use of Lie algebras if the domain transformations are described via Lie group, and all the local "linear" information about these transformations is available from corresponding infinitesimal operator?

  2. What is the essence of classification? I.e., what are the key features of differential equations used to distinguish one type from another? Why are they important?

  • $\begingroup$ As I explained in the summary of an edit on another question of yours a couple of minutes ago (so you might not have seen it), the (analysis) tag is somewhat deprecated and should not be used when there are more specific tags. $\endgroup$ – epimorphic May 17 '15 at 6:01
  • $\begingroup$ @epimorphic Thank you for pointing it out. I was using it in combination with the "differential equations" tag in an attempt to distinguish my "analysis of PDEs" questions from the general (mostly applied) PDE-related topics. I will avoid doing so from now on. $\endgroup$ – Vlad May 17 '15 at 6:05
  • $\begingroup$ That actually sounds pretty reasonable – perhaps a case can be made for its use on questions about differential equations after all. (Though more specific tags with similarly theoretical connotations such as real analysis or regularity theory should probably still be preferred if applicable.) $\endgroup$ – epimorphic May 17 '15 at 6:58
  • $\begingroup$ @Vlad Could you please post some references to the "lot of articles" you mention "related to 'classification of differential equations via finding corresponding Lie algebras' [that] are published every year."? thanks $\endgroup$ – Geremia Jan 3 '17 at 15:39

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