By Charles G. Moore

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Iii) No further relations of order k or less can be derived from E by diﬀerentiation or taking compatibility conditions. 12. A regular system of diﬀerential equations is locally solvable. ’s. e. 11 can be checked by a ﬁnite algorithm [56], and this makes the criterion of regularity useful in practice. If condition (iii) above fails, the Janet theory [33], or Reid’s variant thereof [56] gives an algorithmic procedure for appending compatibility conditions until the condition is satisﬁed. This is often of no concern.

Then Q ˆ consists of equivalence transforfor every augmented operator X mations of C. ˆ of the group Q ˆ satisfy the conditions of the theorem. Proof. 39) shows that Q of the auxiliary system A. 40) is identical to ˆf = 0 X ˆg = 0 X ·κ kκ when f = 0 and g = 0. 13 to the surface E∩A : f = 0, g = 0 shows that Q of transformations leaving invariant E ∩ A. 3 shows that such transformations are equivalence transformations. 40) the inﬁnitesimal We call the set of operators X (augmented) equivalence group for the class C of equations.

3. 41). Since equivalence transformations form a group, the inﬁnitesimal equivalence ˆ form a Lie algebra L ˆ of operators on the space (x, u, a). 40) the Lie algebra of equivalence operators for the class C of equations. 5. 41) discussed above. 1. The Lie algebra of operators ˆ 5, X ˆ 6, X ˆ 7, X ˆ 8 } which actually aﬀect D(u) appears in the lower right hand ˆ = {X R ˆ where K ˆ = ˆ = K ˆ ⊕s R, corner. Note that the algebra is a semidirect sum L ˆ ˆ ˆ ˆ {X1 , X2 , X3 , X4 } is the algebra of operators which do not aﬀect D(u).