A system of the Burgers equations of the two-velocity hydrodynamics is obtained. We consider the Cauchy problem in the case of a one-dimensional system. The estimate of the stability of the solution is obtained. We have obtained a formula for the Cauchy problem for the one-dimensional system of equations that arises in the two-velocity hydrodynamics. It is shown that with disappearance of the kinetic friction coefficient, which is responsible for the energy dissipation, this formula turns to the famous Cauchy problem for the one-dimensional Burgers equation. The existence and uniqueness of solutions to the Cauchy problem for the one-dimensional systems of the Burgers type are proved using the method of weak approximation.