The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

Assuming the source of energy momentum tensor as perfect fluid, a classification of static cylindrically symmetric spacetimes in f (R) theory of gravity by conformal vector fields (CVFs) is presented. For the classification purpose, we put different conditions on metric coefficients to obtain solutions in f (R) theory of gravity. By means of some algebraic and direct integration techniques, it is shown that the dimension of CVFs for the considered spacetimes turns out to be 4, 5, or 15.Mathematical Subject Classification 83C05 · 83C15 · 83C20

We first find the dust solutions of static plane symmetric spacetimes in the theory of f(R) gravity. Then using the direct integration technique on the solutions obtained, we deduce the conformal vector fields. This is performed in the context of f(R) theory of gravity. There exist six cases. Out of these, in five cases the spacetimes become conformally flat and admit 15 conformal vector fields, whereas in the sixth case, conformal vector fields become Killing vector fields.