  In the context of the composite boson interpretation, we construct the exact
general solution of the Dirac--K\"ahler equation for the case of the spherical
Riemann space of constant positive curvature, for which due to the geometry
itself one may expect to have a discrete energy spectrum. In the case of the
minimal value of the total angular momentum, $j=0$, the radial equations are
reduced to second-order ordinary differential equations, which are
straightforwardly solved in terms of the hypergeometric functions. For non-zero
values of the total angular momentum, however, the radial equations are reduced
to a pair of complicated fourth-order differential equations. Employing the
factorization approach, we derive the general solution of these equations
involving four independent fundamental solutions written in terms of
combinations of the hypergeometric functions. The corresponding discrete energy
spectrum is then determined via termination of the involved hypergeometric
series, resulting in quasi-polynomial wave-functions. The constructed solutions
lead to notable observations when compared with those for the ordinary Dirac
particle. The energy spectrum for the Dirac-K\"ahler particle in spherical
space is much more complicated. Its structure substantially differs from that
for the Dirac particle since it consists of two paralleled energy level series
each of which is twofold degenerate. Besides, none of the two separate series
coincides with the series for the Dirac particle. Thus, the Dirac--K\"ahler
field cannot be interpreted as a system of four Dirac fermions. Additional
arguments supporting this conclusion are discussed.
