Exterior differential systems on Lie algebroids and the invariant inverse problem of the calculus of variations

The paper’s Theorem 1 states exactly the equivalence between the existence of a reduced multiplier matrix k_ij and a Lie–algebroid 2-form Ω of maximal rank with Ω(Wi,Wj)=0, ι_{Γ0}Ω=0, and δΩ=0, on the IP Lie algebroid with adapted frame {Γ0,Wi,Hi} . The paper proves that any such Ω must be of the form Ω = k_ij Ψ^i∧Θ^j (no H∧H term) via L_{Γ0}Ω=0 (using ι_{Γ0}Ω=0 and δΩ=0) , and computes δΩ on the triples (Γ0,Wi,Wj), (Γ0,Wi,Hj), (Γ0,Hi,Hj), (Wi,Wj,Hk) to recover precisely the reduced Helmholtz conditions (symmetry, dynamical equation in λ, φ-compatibility, and integrability) , with the remaining δΩ(Wi,Hj,Hk) and δΩ(Hi,Hj,Hk) implications shown to be redundant consequences (eqs. (27)–(28)) . The candidate solution reproduces the same adapted-frame brackets (Lemma 1) and dual-coframe exterior derivatives used in the paper , defines Ω = k_ij W^i∧H^j, and checks δΩ on the same families of triples to derive the same conditions. One minor imprecision is the claim that, from Ω(W,W)=0 and ι_{Γ0}Ω=0 alone, “the only possibly nonzero pairings … involve one W and one H”; the paper rightly uses L_{Γ0}Ω=0 (hence δΩ=0 as well) to eliminate a possible H∧H component before concluding Ω has only W–H blocks . Aside from this small clarity issue, the reasoning and structure match the paper’s proof.