The equivalence of the Haar system in a rearrangementinvariant space $ X $ on $ [0,1] $ and a sequence of pairwise disjoint functionsin some Lorentz space is known to imply that $ X=L_{2}[0,1] $ up to the equivalence ofnorms. We show that the same holds for the class of uniformdisjointly homogeneous rearrangement invariant spaces and obtain a fewconsequences for the properties of isomorphic embeddings of such spaces.In particular, the $ L_{p}[0,1] $ space with $ 1<p<\infty $ is theonly uniform $ p $ -disjointly homogeneous rearrangement invariant space on $ [0,1] $ with nontrivial Boyd indices which has two rearrangement invariant representationson the half-axis $ (0,\infty) $ .
