Adjoint symmetries for graded vector fields

Suppose that ${\mathcal{M}}=(M,{\mathcal A}_M)$ is a graded manifold and consider a direct subsheaf $\cd$ of ${ Der {\mathcal A}_M}$ and a graded vector field $\Gamma$ on ${\mathcal{M}}$, both satisfying certain conditions. $\cd$ is used to characterize the local expression of $\Gamma$. Thus we review some of the basic definitions, properties, and geometric structures related to the theory of adjoint symmetries on a graded manifold and discuss some ideas from Lagrangian supermechanics in an informal fashion. In the special case where ${\mathcal{M}}$ is the tangent supermanifold, we are able to find a generalization of the adjoint symmetry method for time-dependent second-order equations to the graded case. Finally, the relationship between adjoint symmetries of $\Gamma$ and Lagrangians is studied.