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We present a procedure which allows one to integrate explicitly the class ofcheckerboard IC-nets which has recently been introduced as a generalisation ofincircular (IC) nets. The latter class of privileged congruences of lines inthe plane is known to admit a great variety of geometric properties which arealso present in the case of checkerboard IC-nets. The parametrisation obtainedin this manner is reminiscent of that associated with elliptic billiards.Connections with discrete confocal coordinate systems and the fundamental QRTmaps of integrable systems theory are made. The formalism developed in thispaper is based on the existence of underlying pencils of conics and quadricswhich is exploited in a Laguerre geometric setting.

Checkerboard incircular nets. Laguerre geometry and parametrisation