Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.
A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.
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