  We propose a model for local dynamics of a perturbed convex real-analytic
Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m,
m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a
Liouville-integrable system of $n$ non-linear rotators via a small analytic
potential. The global bifurcation problem is set-up for the $n$-dimensional
isotropic manifold, corresponding to a specific homoclinic orbit of the
toroidal pendulum. The splitting of this manifold can be described by a scalar
function on an $n$-torus, whose $k$th Fourier coefficient satisfies the
estimate $$O(e^{- \rho|k\cdot\omega| - |k|\sigma}), k\in\Z^n\setminus\{0\},$$
where $\omega\in\R^n$ is a Diophantine rotation vector of the system of
rotators; $\rho\in(0,{\pi\over2})$ and $\sigma>0$ are the analyticity
parameters built into the model. The estimate, under suitable assumptions would
generalize to a general multiple resonance normal form of a convex analytic
Liouville integrable Hamiltonian system, perturbed by $O(\eps)$, in which case
$\omega_j\sim\omeps, j=1,...,n.$
