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By analytically continuing the eigenvalue problem of a system of two coupledharmonic oscillators in the complex coupling constant $g$, we have found acontinuation structure through which the conventional ground state of thedecoupled system is connected to three other lower {\it unconventional} groundstates that describe the different combinations of the two constituentoscillators, taking all possible spectral phases of these oscillators intoaccount. In this work we calculate the connecting structures for the higherexcitation states of the system and argue that - in contrast to the four-foldRiemann surface identified for the ground state - the general structure iseight-fold instead. Furthermore we show that this structure in principleremains valid for equal oscillator frequencies as well and comment on thesimilarity of the connection structure to that of the single complex harmonicoscillator.

Analytic eigenvalue structure of a coupled oscillator system beyond the ground state