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.
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?
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?