On the theory of Lie–Bäcklund transformation groups

The attempt to apply the theory of Lie groups to the case of Lie–Bäcklund transformations, which are certain tangent transformations of infinite degree, leads to an infinite-dimensional analogue of Lie's equations. This constitutes the main difficulty in any attempt to construct an analytic theory of Lie–Bäcklund transformation groups. In this paper an algebraic solution of this difficulty by means of power series is suggested. A formal theory which preserves the principal features of Lie's theory of tangent transformations is constructed. Some applications of this theory to the group theoretic study of differential equations in which the use of Lie–Bäcklund transformations is essential are considered.
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