An isometry of a Riemannian manifold is an infinitesimal displacement generated by a Killing vector field $V=\zeta^aV_a=\zeta^aV_a^i\frac{\partial}{\partial x^i}$. If the isometry corresponds to the action of a Lie group $G$ on the manifold, then $V_a$ gives rise to the corresponding Lie algebra $\mathfrak{g}$, i.e.,
$[V_a,V_b]^i=-if_{ab}^{c}V_c^i$
(To be precise, we have an antihomomorphism from the Lie algebra to the relevant Killing vector fields, hence the negative sign on the RHS.)
My question is whether there exist manifolds with affine Kac-Moody group symmetry, with the relevant Killing vector fields giving rise to an affine Kac-Moody algebra:
$[V_{an},V_{bm}]^i=-if_{ab}^{c}V_{c(m+n)}^i-c^i(x)m\delta_{m+n,0}\delta_{ab}$
Here, $c(x)$ is a vector field which represents the central element of the affine algebra.
If there are such manifolds, what functional form would the Killing vector fields have?