In the work of J. K. Knowles and E. Sternberg some classes of conservation laws in linear and finite elastostatics are derived using Neter's theorem. In the same paper, it was shown that the corresponding coordinate and vector transformations are unique in the case of linear elastostatic states. The proof of uniqueness was performed under the assumption that Lame's constant $\lambda=0$, which, even in that narrow special case, further narrows the scope of the above-mentioned transformations and leads to the doubt that there are other transformations to which the new conservation laws correspond. In this paper, it is shown that the coordinate and vector transformations performed in the mentioned paper are unique and under more general assumptions that are physically justified. The proof itself is simpler, reviewed and can serve as a method for proving the (non)uniqueness of transformations in the case of nonlinear elasticity.