An oriented triple system on a v-set X is called an HTS (v, λ) if it contains both cyclic and transitive triples such that each ordered pair of distinct elements is contained in exactly λ triples of If all the cyclic and transitive triples from X can be partitioned into U, such that each (X) is an HTS (v,λ), then {(X,)}, is called an LHTS (v,λ). In this paper, the existence spectrum of LHTS(v, 2) is completed, that is 4(v-2)=0 (mod 2), 4 (v-2)≥λ, if λ≠0(mod 3) then v≠2(mod 3) and if λ=1 then v≠3.