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Abstract: After briefly reviewing the well-known necessary and sufficient existence conditions for a nonlinear observer with linear error dynamics (NOLED), we show how spline functions can be used to solve the generalized characteristic equation (GCE). The splines transform the GCE, which is a high-order partial differential equation, into a well-conditioned algebraic one. This equation can be solved even if the original GCE does not posses a solution, i.e., when the non-generic involutivity condition is not satisfied. In this case, we compute an observer with approximately linear error dynamics. Moreover, our approximation is optimal in a well-defined sense. Two examples, one of which is not integrable, illustrate observer construction and performance.