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Abstract: Nonlinear systems admitting observers with linear error dynamics, denoted integrable systems, rarely arise in practice. However, any nonlinear system can be expressed as the sum of non-integrable and integrable parts. We find coordinates wherein the derivative of the system's non-integrable part is uniformly minimized. Using these coordinates, we design a constant gain observer with approximately linear, locally stable error dynamics. Comparison with a Luenberger-like constant gain observer shows our observer yields a larger region of attraction.