Linear invariant relations of Kirchhoff's equations

A description of all of the cases of the existence of a linear invariant relation of Kirchhoff's equations is obtained. Existence conditions in a parametric vector form and in an explicit coordinate form are presented. New linear invariant relations and partial solutions are found. The inclusion of well-known linear invariant relations in the general case obtained is indicated. A geometrical interpretation of the conditions for the Chaplygin linear invariant relation to exist is given.