  In this paper, we study the Cauchy problem for an integrable multi-component
(2N-component) peakon system which is involved in an arbitrary polynomial
function. Based on a generalized Ovsyannikov type theorem, we first prove the
existence and uniqueness of solutions for the system in the Gevrey-Sobolev
spaces with the lower bound of the lifespan. Then we show the continuity of the
data-to-solution map for the system. Furthermore, by introducing a family of
continuous diffeomorphisms of a line and utilizing the fine structure of the
system, we demonstrate the system exhibits unique continuation.
