Please, how do I go about showing that the first integrals of the following n-th order differential equation:
$$ \frac{d^n u}{dx^n} = H(x, u^{n-1})~~ $$
on $M\subset X \times U \simeq\mathbf{R}^2 $are the same as the invariants of the one-parameter group generated by
$$ \partial _x + u_x \partial_u + u_{xx}\partial_{u_x}+u_{xxx}\partial_{u_{xx}}+ \ldots + u_{n-1}\partial_{u_{n-2}}+H(x,u^{n-1})\partial_{u_{n-1}} $$
acting on the jet space $M^{n-1}$.
$U$ is the space of dependent variables, $M$ is an open subset of the space of independent and dependent variables. $X$ is the space of independent variables.
Thank you.