Journal of Geometry and Physics 42 (2002) 195–215 Explicit solutions of integrable lattices Manuel Mañas a,∗ , Luis Mart´ ınez Alonso a , Elena Medina b a Departamento de F´ ısica Teórica II, Universidad Complutense, E28040 Madrid, Spain b Departamento de Matemáticas, Universidad de Cádiz, E11510 Puerto Real, Cádiz, Spain Received 30 January 2001 Abstract Explicit examples of quadrilateral lattices and their integrable reductions of pseudo-circular, symmetric and pseudo-Egorov types are presented. © 2002 Elsevier Science B.V. All rights reserved. MSC: 58B20 Subj. Class.: Dynamical systems Keywords: Integrable lattices; Solutions 1. Introduction This paper focuses its attention on the integrable aspects of discrete geometry [2]. Our main result is the construction of explicit families of quadrilateral, pseudo-circular, symmet- ric and pseudo-Egorov lattices by applying particular fundamental transformations [9,12] to the Cartesian lattice. This particular choice is suggested by previous papers [6,8,13,15,16] in which the Cauchy propagator [19] was extensively used in the study of integrable lattices and nets. In fact, our matrix function D(z,z ′ ) introduced below can be understood as the Cauchy propagator of a particular Cartesian lattice and our fundamental transformations as dressing transformations of it. The advantage of this D(z,z ′ ) compared to that used in, for example, [6,8] is that the reductions follow the same patron as in the continuous case and the ¯ ∂ reduction theory simplifies (private communication by L. Bogdanov). As the solutions obtained in this paper are produced by applying fundamental transfor- mations, one should expect some N -dimensional discrete integration in order to find the transformation potentials. However, this is not the case, and only complex integration is ∗ Corresponding author. E-mail addresses: manuel@darboux.fis.ucm.es (M. Mañas), luism@eucmos.sim.ucm.es (L. Mart´ ınez Alonso), elena.medina@uca.es (E. Medina). 0393-0440/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0393-0440(01)00085-7