Lie–Hamilton systems associated with the symplectic Lie algebra sp(6,R)

The paper correctly constructs the sp(6,R) Lie–Hamilton system on T*R^3, lists the 21 quadratic Hamiltonians h_i, and gives the quadratic Casimir C2; from this it derives F(1)=0 and an explicit two-copy invariant F(2)=−(p(2)·q(1)−p(1)·q(2)+p(2)·q(1)−… )^2, i.e., the negative square of the symplectic pairing between two copies. It also states that the minimal number of copies for a superposition rule is s=6, as the diagonal prolongations become generically independent at that level . However, when formulating the system of algebraic equations purported to yield the superposition rule, the paper chooses six invariants F(2)=−k1^2 and F(2)_{23},F(2)_{24},F(2)_{25},F(2)_{26},F(2)_{27}=−k_ℓ^2; of these, only F(2)_{27} actually involves the unknown copy (7), so the stated system cannot determine z_7. This appears to be a typographical/selection error, and the explicit superposition rule is then omitted as “cumbersome” . The model’s solution reproduces the core paper results (closure of quadratic Hamiltonians and identification with sp(6,R); the C2-based constants; F(1)=0 and F(2)=−(z_1^T J z_2)^2), and then provides a clean, correct superposition rule by choosing the six invariants that each couple the unknown with one known solution, F(2)_{i7}=−k_i^2 (i=1,…,6), yielding a linear system R(t) z_7=κ with R’s rows (−p^(i), q^(i)) and the explicit formula z_7=R^{-1} κ on the open set where det R≠0. This fixes the paper’s omission and the apparent mis-selection of invariants, while making the construction explicit. The only added assumption (generic invertibility of R) is standard and aligns with the genericity invoked by s=6; it should be stated explicitly.