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We discuss the Tzitzeica equation and the spectral properties associated with its Lax operator L. We prove that the continuous spectrum of L is rotated with respect to the contour of the Riemann-Hilbert problem with angle π/6. We also show that the poles of the dressing factors and their inverses are discrete eigenvalues of L. [ABSTRACT FROM AUTHOR]

Abstract: The paper is devoted to the Lie symmetries of  nonlinear dynamical systems described by second order partial differential equations. By imposing the invariance condition of the equations under the action of the Lie symmetry operator we obtained a determining system which could be solved in two directions: (i) to find the symmetries of a concrete equation; (ii) to obtain all the compatible equations with an imposed form of symmetry algebra. This paper will pay attention to the indirect problem (ii) and will use the algorithm for determining the most general equations which belong, following their symmetries, to the same class with two interesting and important physical models: the nonlinear heat equation and the transfer equation with power-law nonlinearities. [Copyright &y& Elsevier]