The Roy equations, combined with unitarity, can be regarded as a system of integral equations for the π-π scattering amplitude in a finite energy region. Even when the partial-wave absorptive parts above this finite range are prescribed, and the two S-wave scattering-length parameters are held fixed, the singular equations have multiple solutions, some of which could be missed in a direct numerical study. We regularize the system by a modified ND method, in which the full manifold of solutions is parametrized explicitly. If δ(s0) is the phase shift of a particular wave at the eutoff point, then that wave carries a number of arbitrary, real parameters equal to the integer part of 2δ(s0)π, provided δ(s0)≥−π2. We suggest that the ND formulation is appropriate for applications of the Roy equations.