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We introduce a new discrete system that arises from ellipsoidal billiards andis closely related to the double reflection nets. The system is defined on thelattice of a uniform honeycomb consisting of rectified hypercubes and crosspolytopes. In the $2$-dimensional case, the lattice is regular and itincorporates dynamics both in the original space and its dual. In the$3$-dimensional case, the lattice consists of tetrahedra and cuboctahedra.

Integrable lattices of hyperplanes related to billiards within confocal quadrics