  The hypothesis of an alternative way of obtaining gravitational waves is the
physical motivation of this article. Using the linear field approximation and a
symmetry transformation of the affine connection, new field equations and new
gauge conditions have been obtained. Solutions to these field equations have
been considered in the empty space and in the wave zone, and in both of them
the oscillation amplitudes of their solutions are attenuated exponentially. We
expect that these solutions can be useful for building more sophisticated
gravitational wave models, and also as an impulse for researching further
symmetry transformations of general relativity.
