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Bäcklund transformations are defined as operations on solutions of a Riemann boundary value problem (vector bundles over P(1)) that add apparent singularities. For solutions of difference and differential linear spectral problems, Bäcklund transformations are presented in explicit form through the Christoffel formula and its generalizations. Identities satisfied by iterations of elementary Bäcklund transformations are represented in the form of the law of addition or as the three-dimensional difference equation of Hirota's type. Matrix two-dimensional isospectral deformation equations are imbedded into three-dimensional scalar systems of Kadomtzev-Petviashvili (law of addition) form. Two-dimensional matrix systems correspond to reductions of Kadomtzev-Petviashvili equations with pseudodifferential operators satisfying algebraic equations.

Laws of composition of Bäcklund transformations and the universal form of completely integrable systems in dimensions two and three