At this stage, we leave the general formulation and adopt the particular choice of K$_{1}$ , K$_{2}$ , G$_{1}$ , and G$_{2}$ given in Eq. (20). The antisymmetric BKNH tensor becomes and the Hamiltonian simplifies to The split Liouville operators are now For the purposes of defining an efficient integration algorithm, we combine commuting Li- 14 ˜ B BKNH =       0 0 0 0 1 0 − q 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 − 1 0 0 0 0 − p 0 0 0 − 1 0 0 p 0 0 − p$_{ζ}$ q 0 − 1 0 0 0 0 − p$_{ξ}$ 0 0 0 − 1 0 p$_{ζ}$ p$_{ξ}$ 0       , (34) ˜ H BKNH = H ( q, p ) + p 2 ζ 2 m$_{ζ}$ + p 2 ξ 2 m$_{ξ}$ + p 2 η 2 m$_{η}$ + k$_{B}$T ( ζ + ξ ) + 2$_{B}$ T η . (35) ˜ L BKNH 1 = − ∂V ∂q ∂ ∂p + q ∂V ∂q ∂ ∂p$_{ξ}$ (36a) ˜ L BKNH 2 = p m ∂ ∂q + p 2 m ∂ ∂p$_{ζ}$ (36b) ˜ L BKNH 3 = − k$_{B}$T ∂ ∂p$_{ζ}$ (36c) ˜ L BKNH 4 = − k$_{B}$T ∂ ∂p$_{ξ}$ (36d) ˜ L BKNH 5 = p$_{ζ}$ m$_{ζ}$ ∂ ∂ζ − p$_{ζ}$ m$_{ζ}$ p ∂ ∂p + p 2 ζ m$_{ζ}$ ∂ ∂p$_{η}$ (36e) ˜ L BKNH 6 = − p$_{ξ}$ m$_{ξ}$ q ∂ ∂q + p$_{ξ}$ m$_{ξ}$ ∂ ∂ξ + p 2 ξ m$_{ξ}$ ∂ ∂p$_{η}$ (36f) ˜ L BKNH 7 = p$_{η}$ m$_{η}$ ∂ ∂η − p$_{η}$ m$_{η}$ p$_{ζ}$ ∂ ∂p$_{ζ}$ − p$_{η}$ m$_{η}$ p$_{ξ}$ ∂ ∂p$_{ξ}$ (36g) ˜ L BKNH 8 = − 2 k$_{B}$T ∂ ∂p$_{η}$ . (36h)