We apply the group theoretical concepts of the EIBM, an extension of the interacting boson model where a distinction is made between particle and hole bosons, to the particular situation of two coexisting systems in the same nucleus exhibiting the O(6) (regular states) and SU(3) (intruder states) symmetries. Besides giving evidence that such a coexisting set of states could well be applicable to the Pb region, we discuss the ''global'' symmetry properties of the above model. We also concentrate on mixing between the otherwise coexisting symmetries and point out how to make more realistic calculations. We discuss, at large, the experimental evidence in the Pb region for intruder analog multiplets. (C) 1997 Elsevier Science B.V.