Vtyjkyuk

I am reading Arnold's ODE but I cannot solve this problem.

This is on page 79.

$mathbfProblem.$ Suppose a one-parameter group of symmetries of a direction field in an $n$-dimensional domain is known. Reduce the problem of integrating the corresponding differential equation to finding the integral curves of a direction field on a domain of dimension $n-1$.

This problem comes with a hint, which I don't know how to prove either.

Hint: The space of orbits of the symmetry group has dimension $n-1$

$mathbfQ1.$ How to prove the hint ?

My thought on Q1.

The symmetry $mathbfg$ is a differmorphism such that $mathbfg'(x)v(x)=mathbfv(g(x))$.

It's a PDE of $n$-dimension but solving $mathbfg(x)$ explicitly does not look promising to prove the hint.

$mathbfQ2.$ If the statement in the hint is true, then how to reduce the ODE by one dimension?

My thought on Q2.

Locally, $mathbfg(x)$ acts as translation so $mathbfg'(x)v(x)=mathbfv(g(x))$ will go like $mathbfv(x)=mathbfv(x+dx)$.

If we have $n-1$ dimension local translation, the direction field will only depend on one coordinate. The original problem will reduce to integrate the curve only on one coordinate and not on $n-1$ dimension locally.

Please help.