Abstract <p>The problem of determining the replacement of independent variables in the partial differential equations of three-dimensional problem of the perfect plasticity theory (for the stress states corresponding to an edge of the Tresca prism) is considered in order to reduce these equations to the analytically simplest Cauchy normal form. The original system of equations is presented in the isostatic coordinate net and is essentially nonlinear. A criterion of maximum simplicity is formulated for the Cauchy normal form. The coordinate net is found to reduce the original system to the analytically simplest Cauchy normal form. The obtained condition when the system of equations takes the simplest normal form, is stronger than the <i>t</i>-hyperbolicity condition of Petrovskii if we take <i>t</i> as the canonical isostatic coordinate which level surfaces form the spatial layers that are normal to the field of the principal directions corresponding to the greatest (or lowest) principal stress.</p>