We present a comprehensive analytical study of a variation of the eigenvector ensemble initially proposed by Deutsch [1, 2] for the foundations of the Eigenstate Thermalization Hypothesis (ETH). This ensemble, called the C-ensemble, incorporates additional system-dependent information, enabling the study of complex quantum systems beyond the universal predictions of Random Matrix Theory (RMT). Specifically, we focus on how system-specific details influence late-time behavior in correlation functions, such as the spectral form factor, and how explicit Hamiltonian corrections not captured by RMT can be included. We demonstrate the consistency of this ensemble with respect to the universal (Haar) results by showing that it defines a unitary 1-design for arbitrary systems and for strongly chaotic systems it becomes an approximated 2-design. Universal expressions for two- and four-point ensemble-averaged correlation functions are derived, revealing how system-dependent information is spectrally decoupled. Furthermore, we show that for small energy windows, the correlation functions defined by this ensemble reduce to the predictions made by the ETH.