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 J n − 1 M , the vector fields μ Γ 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 J n − 1 M with first-order λ -prolongation Y and first-order standard prolongations Z of X defined on J n M , we prove that g Y cannot be first-order λ -prolonged vector field of vector field g X 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 − 1 th 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 X i , i = 1 , . . . , n that are in involution, We claim that under some conditions, their λ -prolongations also in involution.