Semi-Hamiltonian integrable hydrodynamic type systems which describe critical points of functions obeying the linear Darboux systems are discussed. It is shown that a wide class of such systems can be constructed using the Lauricella type solutions of the Euler-Poisson-Darboux equations as sead functions. Classical multi-phase Whitham equations for the Korteweg- de Vries and nonlinear Schrodinger equations are the particular examples of such systems.