  We present an exhaustive analytical exploration of a variation of the
eigenvector ensemble initially proposed by Deutsch for the foundations of the
Eigenstate Thermalization Hypothesis (ETH) that drastically (but not fully)
increases system-dependent information, and therefore serves as a model for the
study of complex quantum systems beyond the universal Random Matrix Theory
(RMT) regime. We especially focus on the remnants of system-specific
information that modify the late-time behavior of correlation functions like
the spectral form factor, or the inclusion of explicit system-dependent
Hamiltonian corrections not captured by the RMT regime. The consistency of the
predictions in the new ensemble with respect to the universal (Haar) one is
made explicit by showing how that for arbitrary systems it defines an unitary
1-design and for strongly chaotic systems it becomes an approximated 2-design.
While we are able to provide universal expressions for two- and four-point
ensemble-averaged correlation functions and show how system-dependent
information is spectrally decoupled, we also show that for small energy windows
correlation functions defined by the new ensemble reduce to the predictions of
the ETH.
