  With respect to earlier investigations, the theory of multi-component,
concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and
extended, including subsystems with nonzero density on the boundary and
subsystems with intersecting boundaries. The formulation is restricted to two
subsystems for simplicity but, in principle, can be extended to $N$ subsystems.
Equilibrium configurations are independent of the nature of the fluid i.e.
collisional or collisionless, provided the polytropic index lies within the
range, $1/2\le n\le5$, as in one-component systems. The solution of the
equilibrium equations is expanded in power series, which can be continued up to
the boundary and outside via starting points placed at increasingly larger
distance from the centre of mass. A detailed analysis is devoted to special
cases where the solution of the equilibrium equations can be expressed
analytically. Finally a guidance example is shown, involving homogeneous
subsystems with intersecting boundaries, where a substantially flattened
component extends outside a slightly flattened one.
