How to generate a symmetry group with a vector field?

Question

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.

Answer 1

See: https://en.wikipedia.org/wiki/Lie_point_symmetry

There another good book about symmetries of DE:

Applications of Lie Groups to Differential Equations, Peter J. Olver.

First, there is a correspondence between Lie algebra consists of vector fields and Lie group. Roughly speaking, Lie algebra $\rightarrow$ Lie group is a integration, while the inverse is differentation about $\epsilon$ and take $\epsilon=0$.

Second, you need to judge whether a group is symmetry of DE. Sind the first sentence, you can judge just from the vector field. (see chapter 2 of Olver's book, there is a theorem)

Besides, the equation in your question is a special one. This is equivalent to a vector field acting on $u$, so the conclusion looks different, but it can be obtained from the general case.