Integrability of Hamiltonian systems on COLCS Manifold
We develop an integrability framework for Hamiltonian dynamics on \emph{con-locally conformal symplectic} (COLCS) manifolds, odd-dimensional quadruples $(M,\Omega,\theta,\eta)$ where $d\Omega=\theta\wedge\Omega$, a closed $1$-form $\eta$ determines a codimension-one distribution on which $\Omega$ is non-degenerate, and $R$ is the Reeb vector field satisfying $\iota_R\Omega=\iota_R\theta=0$, $\iota_R\eta=1$. This class simultaneously generalises LCS and cosymplectic manifolds and provides a natural arena for time-dependent Hamiltonian systems with twisted differential $d^\theta=d-\theta\wedge$. The COLCS bracket is introduced on $C^\infty(M)$ and shown to be a Lie bracket that induces Poisson structures on the subalgebras of $\theta$-strong ($\theta(X_H)=0$) and $R$-strong ($R(H)=0$) functions. A Lie-type integrability theorem is then established: given $2n-k$ functionally independent first integrals with $k$ of them $\theta$-strong and generating a solvable Lie algebra under the COLCS bracket, the flow is integrable by quadratures on the common level set. Finally, scaling symmetries of degree $(\Lambda,\beta,\gamma)$, defined by $L_X\Omega=\beta\Omega$, $L_XH=\Lambda H$, $L_X\eta=\gamma\eta$, are studied: they rescale $X_H$ by $(\Lambda-\beta)$, generate families of first integrals, and imply structural primitives for $\Omega$ and $\eta$.