I'm trying to study Lie groups applied to solution of differential equations, and I'm working with the following problem
Consider a first order homogeneous linear partial differential equation $$\sum_{i=1}^p\xi^i(x)\frac{\partial u}{\partial x^i}=0,\tag{$\ast$}$$ and let $v=\sum\xi^i(x)\partial_i$ be the corresponding vector field.
Show that $w=\sum\eta^i(x)\partial_i$ generates a one-parameter symmetry group if and only if $[v,w]=\gamma v$ for some scalar-valued function $\gamma(x)$.
I'm having trouble understanding the part where $w$ generates the symmetry group, how do i usually proceed in this case?
How do you generate a Group with the vector space $w$?
Since, at least what i have seen so far, when given a local group action like $\phi(\epsilon,(x,y))=\left(\frac{x}{1-\epsilon x},\frac{y}{1-\epsilon x}\right)$, i know how to get the infinitesimal generators and how to check if it form a one-parameter Lie group of symmetries of a ODE given, but I don't get how from a vector field $w$ one can generate a symmetry group.
Any help on what i could study to get this that I'm missing would be appreciated, thanks in advance.