I was also going to comment on CSIC but all mathematicians I've met at ICMAT have a permanent position in an University thus some teaching assignment.

I am not sure because I have read neither of them... yet. I am still with the "soft" introduction by Hydon

After two years' study I am still struggling with Cartan's equivalence method, see e.g. Olver. But if you are looking for a result more or less algorithmic, easy and immediately applicable, I think the search is pretty hopeless.

@marvin Please note that the question remains "unanswered" if the answer is not accepted and/or upvoted

@marvin I am no expert in differential geometry but I guess that, if the system has some symmetry, it can be formulated on a low-dimensional manifold so that, in practice, there are "less" variables and some of them can be neglected.