Identifying Noetherian symmetries from general Lie symmetries of a differential equation

I know the Lie symmetry group of the harmonic oscillator differential equation from the literature. It is an 8 parameter Lie group.

5 of these generators generate Noether symmetry: $$G_1=\sin(2t)\frac{\partial}{\partial t}+q\cos(2t)\frac{\partial}{\partial q}$$ $$G_2=\cos(2t)\frac{\partial}{\partial t}-q\sin(2t)\frac{\partial}{\partial q}$$ $$G_3=\cos(t)\frac{\partial}{\partial q}$$ $$G_4=\sin(t)\frac{\partial}{\partial q}$$ $$G_5=\frac{\partial}{\partial t}$$

The other three are: $$G_6=q\frac{\partial}{\partial q}$$ $$G_7=q\sin(t)\frac{\partial}{\partial t}+q^2\cos(t)\frac{\partial}{\partial q}$$ $$G_8=q\cos(t)\frac{\partial}{\partial t}-q^2\sin(t)\frac{\partial}{\partial q}$$

I want to identify generators of Noether symmetry for other differential equations. I can get generators of entire symmetry groups using Maple. I can also get conserved current, but I need some help in identifying which generators correspond to conserved quantities using Maple or some other software.

New contributor
ilawid is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• I edited your post to write the equations using Mathjax. In the future, you should format your posts in this way, as it makes the equations searchable and readable by those using a screen reader. Nov 23 at 22:07
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Nov 24 at 17:19
• This question is not about computations, it is about physics Nov 24 at 17:42