As well know, any local regular, analytic and holonomic dynamical systems with finite dimensions can be reduced into Birkhoffian systems, which become self-adjoint.The self-adjoint Birkhoffian systems are of significance because they possess a general symplectic structure.Furthermore, Lie algebra is admissible for autonomous Birkhoffian systems.Therefore, Birkhoffian systems supply a symbiosis order of self-adjointness,symplecticity and Lie algebra almost like Hamiltonian systems.