  The question of the uniqueness of energy-momentum tensors in the linearized
general relativity and in the linear massive gravity is analyzed without using
variational techniques. We start from a natural ansatz for the form of the
tensor (for example, that it is a linear combination of the terms quadratic in
the first derivatives), and require it to be conserved as a consequence of
field equations. In the case of the linear gravity in a general gauge we find a
four-parametric system of conserved second-rank tensors which contains a unique
symmetric tensor. This turns out to be the linearized Landau-Lifshitz
pseudotensor employed often in full general relativity. We elucidate the
relation of the four-parametric system to the expression proposed recently by
Butcher et al. "on physical grounds" in harmonic gauge, and we show that the
results coincide in the case of high-frequency waves in vacuum after a suitable
averaging. In the massive gravity we show how one can arrive at the expression
which coincides with the "generalized linear symmetric Landau-Lifshitz" tensor.
However, there exists another uniquely given simpler symmetric tensor which can
be obtained by adding the divergence of a suitable superpotential to the
canonical energy-momentum tensor following from the Fierz-Pauli action. In
contrast to the symmetric tensor derived by the Belinfante procedure which
involves the second derivatives of the field variables, this expression
contains only the field and its first derivatives. It is simpler than the
generalized Landau-Lifshitz tensor but both yield the same total quantities
since they differ by the divergence of a superpotential. We also discuss the
role of the gauge conditions in the proofs of the uniqueness. In the Appendix,
the use of the symbolic tensor manipulation software Cadabra is briefly
described.
