  The first example of the so-called "coupled" integrable hydrodynamic chain is
presented. Infinitely many commuting flows are derived. Compatibility
conditions of the first two of them lead to the remarkable Manakov--Santini
system. Integrability of this four component three dimensional quasilinear
system of the first order as well as the coupled hydrodynamic chain is proved
by the method of hydrodynamic reductions. In comparision with a general case
considered by E.V. Ferapontov and K.R. Khusnutdinova, in this degenerate case N
component hydrodynamic reductions are parameterized by N+M arbitrary functions
of a single variable, where M is a number of branch points of corresponding
Riemann surface. These hydrodynamic reductions also are written as symmetric
hydrodynamic type systems. New classes of particular solutions are found.
