On the Properties of λ-Prolongations and λ-Symmetries

In this paper, (1) We show that if there are not enough symmetries and λ-symmetries, some first integrals can still be obtained. And we give two examples to illustrate this theorem. (2) We prove that when X is a λ-symmetry of differential equation field Γ, by multiplying Γ a function μ defineded on Jn−1M, the vector fields μΓ can pass to quotient manifold Q by a group action of λ-symmetry X. (3) If there are some λ-symmetries of equation considered, we show that the vector fields from their linear combination are symmetries of the equation under some conditions. And if we have vector field X defined on Jn−1M with first-order λ-prolongation Y and first-order standard prolongations Z of X defined on JnM, we prove that gY cannot be first-order λ-prolonged vector field of vector field gX if g is not a constant function. (4) We provide a complete set of functionally independent (n−1) order invariants for V(n−1) which are n−1th prolongation of λ-symmetry of V and get an explicit n−1 order reduced equation of explicit n order ordinary equation considered. (5) Assume there is a set of vector fields Xi,i=1,...,n that are in involution, We claim that under some conditions, their λ-prolongations also in involution.