A dynamic inverse problem for a one-dimensional system of the Hopf-type equations is considered. A theorem on local solvability in the class of functions
analytic in the variable *x* is proved.

A one-dimensional system of the Hopf-type equations is considered. Axial solutions to problems in the field of modeling two-fluid interactions are sought. A nonlinear system of ordinary differential equations is obtained. Direct and inverse problems for the obtained ODE are considered. A theorem on local solvability is proven.

A flow of incompressible viscous two-velocity fluids for the case of pressure equilibrium of phases at constant saturation of substances is described with the help of scalar functions. A system of differential equations for these functions is obtained. An example illustrating this method is presented.

The fundamental solution to describe the three-dimensional steady-state flows of viscous fluids of the two-velocity continuum with pressure phase equilibrium has been obtained.