<p>Achieving ergodicity in three-dimensional conservative chaotic systems remains a formidable challenge, severely restricting their deployment in applications such as cryptography. This research presents a novel methodology for constructing such systems by bypassing the constraints inherent in Boltzmann-Gibbs statistical mechanics. The proposed framework leverages the unique structural properties of matrix differential equations inspired by the Nosé-Hoover thermostat. To demonstrate its efficacy, we designed and systematically analyzed a prototypical system. Through comprehensive numerical simulations and experimental validation, we uncovered a diverse range of complex dynamics, encompassing strong and weak chaos as well as invariant tori. Critically, we establish that the system exhibits robust ergodicity across specific parameter domains. Its chaotic output satisfies the rigorous National Institute of Standards and Technology (NIST) test suite. Consequently, this work introduces a new blueprint for the systematic design of ergodic conservative chaotic systems, thereby creating significant opportunities for the development of next-generation information security solutions based on conservative chaos.</p>