  We study the conditions under which one can conserve local translationally
invariant operators by local translationally invariant Lindblad equations in
one-dimensional rings of spin-1/2 particles. We prove that for any 1-local
operator (e.g., particle density) there exist Lindblad dissipators that
conserve that operator, while on the other hand we prove that among 2-local
operators (e.g., energy density) only trivial ones of the Ising type can be
conserved, while all the other cannot be conserved, neither locally nor
globally, by any 2- or 3-local translationally invariant Lindblad equation. Our
statements hold for rings of any finite length larger than some minimal length
determined by the locality of Lindblad equation. These results show in
particular that conservation of energy density in interacting systems is
fundamentally more difficult than conservation of 1-local quantities.
