  We study the structure of quantum states for a binary mixture of spin-1
atomic Bose-Einstein condensates. In contrast to collision between identical
bosons, the s-wave scattering channel between inter-species does not conform to
a fixed symmetry. The spin-dependent Hamiltonian thus contains non-commuting
terms, making the exact eigenstates more challenging to obtain because they now
depend more generally on both the intra- and inter-species interactions. We
discuss two limiting cases, where the spin-dependent Hamiltonian reduces
respectively to sums of commuting operators. All eigenstates can then be
directly constructed, and they are independent of the detailed interaction
parameters.
