Functions Whose Poisson Brackets Are Constants

A local normal form under canonical transformation is found for n independent functions of 2n variables, with the condition that the Poisson bracket of each pair of the functions be constant. The normal form, closely related to work of Lie, is used to prove a conjecture of Avez on 1-forms in involution and to obtain a criterion for n independent functions of 2n variables to be extendable to a canonical coordinate system. The last result has been obtained in different ways by Lie and Kruskal.