Lie and Noether symmetries in some classes of pp-wave spacetimes

We derive the Lie point symmetries of geodesic equations and and Noether gauge symmetries of the geodesic Lagrangian for some classes of pp-wave spacetimes. The considered classes of pp-wave spacetimes are the plane-wave spacetimes (i) H=αx²u⁻², (ii) H=α²x²u⁻⁴ and (iii) H=α²x², and a pp-wave spacetime (iv) H=αxⁿ, where H is the scale factor in pp-wave spacetime. We discuss the relations between the obtained Lie point and Noether gauge symmetry generators. We also calculate the first integrals in each of the cases, and give a solution of geodesic equations using those first integrals in case (i) as an example. (paper)