INVARIANTS, SOLUTIONS AND INVOLUTION OF HIGHER ORDER DIFFERENTIAL SYSTEMS

Abstract

The paper is concerned with the interpretation of the fixed points of an involution as invariant solutions under certain Lie algebra of symmetries of a given equation. Our aim is to study the involutivity in terms of the symmetries of an equation. We prove that if $\pi :E\to M$ is a fiber bundle and $\mathrm{\nabla }:T^{\ast }M\to J^1{T}^{\ast }M$ is a linear connection on the base space, then there exists a unique involutive linear automorphism, ${\alpha }_{{}_{\mathrm{\nabla }}}$ in $J^1{J}^{1}E$, that commutes with the projections ${\pi }_{11}$ and $J^1{\pi }_{1,0}$. Moreover, we prove that the space $J^k(\pi )$ is the quotient space of the iterated sesqui-holonomics jets $\text{^}{J}^1{J}^{k-1}(\pi )$ relative to the subgroup of symmetries determined by some involution ${\alpha }_g$.