  After introducing the generators and irreducible representations of the ${\rm
su}(5)$ and ${\rm so}(6)$ Lie algebras in terms of the Schwinger's scillators,
the general kernel solutions of the Kostant operators on eight-dimensional
quotient spaces ${\rm su}(5)/{\rm su}(4)\times {\rm u}(1)$ and ${\rm
so}(6)/{\rm so}(4)\times {\rm so}(2)$ are derived in terms of the diagonal
subalgebras ${\rm su}(4)\times {\rm u}(1)$ and ${\rm so}(4)\times {\rm so}(2)$,
respectively.
