  We present a symmetry classification of the linearised Navier-Stokes
equations for a two-dimensional unbounded linear shear flow of an
incompressible fluid. The full set of symmetries is employed to systematically
derive invariant ansatz functions. The symmetry analysis grasps three
approaches. Two of them are existing ones, representing the classical normal
modes and the Kelvin modes, while the third is a novel approach and leads to a
new closed-form solution of traveling modes, showing qualitatively different
behaviour in energetics, shape and kinematics when compared to the classical
approaches. The last modes are energy conserving in the inviscid case. They are
localized in the cross-stream direction and periodic in the streamwise
direction. As for the kinematics, they travel at constant velocity in the
cross-stream direction, whilst in the streamwise direction they are accelerated
by the base flow. In the viscous case, the modes break down due to damping of
high wavenumber contributions.
