  We have found two non-trivial massless Dirac and two massive Rarita-Schwinger
solutions in plane wave spacetimes. The first order symmetry operator
transforming one of the massless Dirac solution to the other is constructed.
The only non-vanishing spinor bilinear generated by the standard spinor basis
is obtained and algebraic relations between the induced parallel forms are
demonstrated. It is also seen that the spin-3/2 norm of the Rarita-Schwinger
solutions enforces to the massless sector.
