I am no expert in solving DEs by symmetry methods, but from pure interest - is it possible for a differential equation to have an infinite-dimesional Lie group as a symmetry group?
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$\begingroup$ This is answered in the review that Igor Rivin suggested in the answer to this past question of yours mathoverflow.net/questions/282450/… . See the discussion after Theorem 4. It also appears that sections 6 and 7 discuss several explicit examples. $\endgroup$– j.c.Commented Oct 7, 2017 at 1:47
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2$\begingroup$ The symmetry group of any linear partial differential equation (that is not an ODE) is infinite dimensional, so the answer is 'yes'. $\endgroup$– Robert BryantCommented Oct 7, 2017 at 13:53
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