  In this paper we introduce a new property of two-dimensional integrable
systems -- existence of infinitely many local three-dimensional conservation
laws for pairs of integrable two-dimensional commuting flows. Infinitely many
three-dimensional local conservation laws for the Korteweg de Vries pair of
commuting flows and for the Benney commuting hydrodynamic chains are
constructed. As a by-product we established a new method for computation of
local conservation laws for three-dimensional integrable systems. The Mikhalev
equation and the dispersionless limit of the Kadomtsev--Petviashvili equation
are investigated. All known local and infinitely many new quasi-local
three-dimensional conservation laws are presented. Also four-dimensional
conservation laws are considered for couples of three-dimensional integrable
quasilinear systems and for triples of corresponding hydrodynamic chains.
