We extend the method of Kazantzis and Kravaris [Systems Control Lett., 34 (1998), pp. 241 247] for the design of an observer to a larger class of nonlinear systems. The extended method is applicable to any real analytic observable nonlinear system. It is based on the solution of a first-order, singular, nonlinear PDE. This solution yields a change of state coordinates which linearizes the error dynamics. Under very general conditions, the existence and uniqueness of the solution is proved. Lyapunov's auxiliary theorem and Siegel's theorem are obtained as corollaries. The technique is constructive and yields a method for constructing approximate solutions.