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For the first time, definite evidence for the existence of two-phonon collective states in deformed nuclei has been established through the measurement of the absolute transition rate for the decay of the double-/ AT*-4 + vibration in 168 Er.PACS numbers: 21.10. Re, 21.60.Ev, 23.20.Ck, 27.70.+q For nearly three decades a significant issue in the nuclear structure of deformed nuclei has been whether or not two-phonon collective excitations, such as, e.g., double-7 or double-/? vibrations, exist. This issue is central to our understanding of collectivity in nuclei and to the role of the Pauli principle in modulating the possible harmonic spectrum of multiphonon vibrational excitations. It also reflects directly on the practical intractability of large-basis shell-model calculations in heavy deformed nuclei and on the consequent needs to truncate the space used in the model calculations. As a consequence of this need, different truncation schemes, introducing different approximations, have reached different conclusions.For example, for years the quasiparticle-phonon nuclear model 1 (QPNM), which restricts the basis to, at most, two-phonon states, has led to the conclusion that two-phonon collective vibrational excitations will not exist in deformed nuclei due to Pauli blocking of important quasiparticle components.On the other hand, the multiphonon method 2 (MPM) embodies an entirely different truncation scheme. It uses only a few collective phonons and restricts the basis to all the corresponding multiphonon states up to eight phonons. This approach predicts that, for strongly collective vibrations, two-phonon K n=z A + excitations should appear at an energy of about 2.6 times the energy of the one-phonon K n = 2 + state and keep some collective character in their decay to that one-phonon state.Another model which does so is the extended interacting-boson model 3 (sdg-\BM) which gives extra SU(3) representations compared to the original sd-l&Nl model. Broken-SU(3) calculations in this approach 3 predict a K K =4 + band near 2 MeV in 168 Er with properties closely resembling a two-phonon y vibration.While these theoretical approaches differ mainly in their truncation schemes, they share an emphasis on the valence shells. By contrast, the dynamic-deformation model 4 (DDM) constructs the collective potential start-ing from a set of deformed single-particle basis states accommodating eight major shells. With no free parameters it also predicts a collective K n==l A + bandhead at almost exactly 2 MeV (but finds 1.075 MeV for the y bandhead).An alternative approach, the pseudosymplectic model (PSM), also brings into play a full multi-fta) space. Here, an algebraic approach 5 utilizing the symplectic group structure Sp(3,7?), based on the pseudo-SU(3) scheme, is exploited which allows the multishell complexity to be reduced by symmetry constraints. The Sp(3,i?) approach, which incorporates several major shells, has the further interesting feature that the giant resonances and low-lying collectivity are treated on the s...