Verify that the 3rd ODE

admit two infinitesimal generator

And using these symmetry to simplify the above ode to a 1rst order ODE.

I could only verify the vector fields indeed are infinitesimal generators of the 3rd ODE. However, I do not know how to simplify the ODE. Would you help me?

This is Problem 6.3 in

A practical course in differential equations and mathematical modelling

by Nail H. Ibragimov,Nailʹ Khaĭrullovich Ibragimov