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In this paper, (1) We show that if there are not enough symmetries and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-symmetries, some first integrals can still be obtained. And we give two examples to illustrate this theorem. (2) We prove that when <i>X</i> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-symmetry of differential equation field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula>, by multiplying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula> a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> defineded on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>M</mi><mo>,</mo></mrow></semantics></math></inline-formula> the vector fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mi mathvariant="normal">Γ</mi></mrow></semantics></math></inline-formula> can pass to quotient manifold <i>Q</i> by a group action of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-symmetry <i>X</i>. (3) If there are some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-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 <i>X</i> defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>M</mi></mrow></semantics></math></inline-formula> with first-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-prolongation <i>Y</i> and first-order standard prolongations <i>Z</i> of <i>X</i> defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>J</mi><mi>n</mi></msup><mi>M</mi><mo>,</mo></mrow></semantics></math></inline-formula> we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>Y</mi></mrow></semantics></math></inline-formula> cannot be first-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-prolonged vector field of vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>X</mi></mrow></semantics></math></inline-formula> if <i>g</i> is not a constant function. (4) We provide a complete set of functionally independent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> order invariants for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>V</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> which are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>th prolongation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-symmetry of <i>V</i> and get an explicit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> order reduced equation of explicit <i>n</i> order ordinary equation considered. (5) Assume there is a set of vector fields <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula> that are in involution, We claim that under some conditions, their <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-prolongations also in involution.