I am reading a paper: https://onlinelibrary.wiley.com/doi/abs/10.1002/oca.2190
For the PDE (6) in that paper: $$V_t + V_{x_1}f_1(x_1,x_2)+V_{x_2}f_2(x_1,x_2)+Q(x_1,x_2)-\frac{b^2}{4r}V^2_{x_2}=0.$$
The paper says that it is invariant under the following Lie point symmetries:
Translation of time $\tilde{t} = t+s$ with group infinitesimal generator $\frac{\partial}{\partial_t}$.
Translation of the value function $\tilde{V} = V+s$ with group infinitesimal generator $\frac{\partial}{\partial V}$.
I really have no idea what this means and how to see this.
Moreover, it says,
the meaning of two symmetries is that there always exists a solution of that PDE that does not depend on time.
Why?
Also are these symmetries helpful in any sense?
Thanks in advance.