  A U(n)-manifold is multiaxial if the isotropy groups are always conjugate to
unitary subgroups. The classification and the concordance of such manifolds
have been studied by Davis, Hsiang and Morgan under much more strict
conditions. We show that in general, without much extra condition, the homotopy
classification of multiaxial manifolds can be split into a direct sum of the
classification of pairs of adjacent strata, which can be computed by the
classical surgery theory. Moreover, we also compute the homotopy classification
for the case of the standard representation sphere. We also present the result
for the similar multiaxial Sp(n)-manifolds.
