Sophus Lie, the mathematician

(F,g) = 0which enters in Jacobi's work on solving (1) becomes for Lie the equation[X 9 ,XF] = 0,which leads to a search for an infinitesimal contact ransformation leavinginvariant the giv'en equation (1). This signals a shift in his worktowards transformation groups and differential equations.After Lie had applied for a professorship in Lund the Norwegianstorting approved a Professorship for him in Christiania, July 1, 1872.He continued his work on contact transformations but then in 1873started on a systematic study of transformation groups. The motivationis the question, stated explicitly in a paper [20] from 1874: "Howcan the knowledge of a stability group for a differential equation be utilizedtowards its integration?" A point transformation is said to leavea differential equation stable if it permutes the solutions. Here Lie is ofcourse inspired by Galois theory for algebraic equations which he hadheard about already in Sylow's lecture in 1862, and had presumablydiscussed with Jordan in Paris, who had just then (1869) publishedhis clarification of Galois' theory. In the quoted paper Lie proves thefollowing now famous theorem, stated below.Considerdy Y(x, y)dx X(x, y)and a local one-parameter subgroupinduced vector fieldt of diffeomorphisms of R 2 witht= t=(p)/ + i ; -J/ay.Theorem. The transformations t leave the equation stable if andonly if the vector field Z = Xa/&z + Yalay satisfies[I, Z] = AZ(A a function).In this case (X7 7 -Y)- is an integrating factor to the equationX dy - Y dx = 0.11