INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 34 (2001) 10413–10421 PII: S0305-4470(01)23941-9 Fundamental transformations for quadrilateral lattices: first potentials and τ -functions, symmetric and pseudo-Egorov reductions Manuel Ma ˜ nas Departamento de F´ ısica Te´ orica II, Universidad Complutense, E28040 Madrid, Spain E-mail: manuel@darboux.fis.ucm.es Received 17 April 2001, in final form 3 September 2001 Published 23 November 2001 Online at stacks.iop.org/JPhysA/34/10413 Abstract We find the behaviour of first potentials and τ -functions for quadrilateral lattices under vectorial fundamental transformations. We also give those transformations which preserve the symmetric and pseudo-Egorov reductions. PACS numbers: 05.50.+q, 02.30.-f 1. Introduction Integrable discrete equations seems to be essential in the understanding of integrable systems. On the one hand many integrable nonlinear PDEs are continuous limits of integrable PEs (partial difference equations). On the other hand many of these integrable nonlinear PDEs are reductions or connected with differential geometry, in particular with the theory of conjugate nets and its reductions. Sometime ago the German geometer Sauer discussed for example discrete pseudo-spherical surfaces. Recently, the group of Bobenko and Pinkall and the group of Doliwa and Santini have given to the theory of integrable lattices an almost closed form. These integrable lattices contain as reductions many of the mentioned discrete integrable systems and constitute a cornerstone in the theory of integrable systems. Among these lattices one finds the quadrilateral, circular, Egorov and asymptotic lattices. In this paper we investigate how the first potentials and τ -functions for quadrilateral lattices transform under fundamental transformations. Using this information we search for those fundamental transformations which reduce to symmetric lattices and pseudo-Egorov lattices. The layout of the paper is as follows: in section 2 we review some well known basic aspects of quadrilateral lattices, reductions and transformations. In section 3 we present the transformation of both first potentials and τ -functions under vectorial fundamental transformations. Finally, section 4 is devoted to characterizing those vectorial fundamental transformations preserving symmetric and pseudo-Egorov lattices. 0305-4470/01/4810413+09$30.00 © 2001 IOP Publishing Ltd Printed in the UK 10413