The evolution equations of a linear viscoelastic solid are written in terms of the Laplace transform of the displacement field. A corresponding reformulation of the condition of vanishing divergence for vector fields is then proposed and, through a systematic procedure, an explicit representation for a very large family of such conserved vectors is derived. As an application it is shown how a suitable choice of the admissible parameters leads to specific conservation laws which involve spatial means of linear momentum, angular momentum, stress, and displacement, in terms of the known body force, and initial and boundary data. As a further application a Betti-type reciprocity relation is derived. The connection with Noether’s approach to conservation laws is also discussed.