In this thesis, conservation laws of the Lame equation describing linear elasticity have been constructed. This has been done by the implementation of two theorems for constructing conservation laws: the new theorem suggested by N H.Ibragimov, and the classical theorem presented by E.Noether. As a result one can see that the Ibragimov method provides more conservation laws than the Noether theorem, which may suggest the better efficiency of the Ibragimov method.