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 $\nabla:T^*M\to J^1T^*M$ is a linear connection on the base space, then there exists a unique involutive linear automorphism, $\alpha_{_{\nabla}}$ in $J^1J^1E$, 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 $\^{J}^1J^{k-1}(\pi)$ relative to the subgroup of symmetries determined by some involution $\alpha_{g}$.