Classification of Three-Dimensional Integrable Scalar Discrete Equations

We classify all integrable three-dimensional scalar discrete affine linear equations Q3 = 0 on an elementary cubic cell of the lattice $${\mathbb Z}^3$$ . An equation Q3 = 0 is called integrable if it may be consistently imposed on all three-dimensional elementary faces of the lattice $${\mathbb Z}^4$$ . Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only non-trivial (non-linearizable) integrable equation from this class is the well-known dBKP-system.

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