<p>A recently developed three-dimensional formalism for the nucleon-deuteron break-up channel, where a nucleon projectile impinges upon a deuteron target and breaks the deuteron up into a proton and a neutron, initially considered only the leading-order term of the Faddeev equations, using the nucleon-nucleon T-matrix to compute the break-up amplitude. The Faddeev equations stem from the Faddeev approach, a theoretical framework designed to study the dynamics of nuclear three-body systems. In the present study, we extend that formalism by solving the full three-nucleon Faddeev equation without three-nucleon-force contributions in a three-dimensional approach in which momentum vectors are used directly as variables. This formalism is well suited for projectile energies in scattering processes above the pion-production threshold, where partial-wave expansions become inefficient. The free three-nucleon propagator contains moving singularities which basically depend on more than one variable. To treat these moving singularities, we introduce a new method that treats these singularities in a manner analogous to the simple poles appearing in the Lippmann–Schwinger equation, which is derived from the well-known Schrödinger equation for a nuclear two-body scattering process. This approach evaluates the moving singularities directly through the center-of-mass energy of the so-called 23-subsystem, comprising the two nucleons labeled 2 and 3, and eliminates the need to partition the domain of the Jacobi momentum q into intervals. In this context. the Jacobi momentum q represents the relative momentum between the projectile nucleon (labeled nucleon 1) and that of the pair of nucleons (labeled nucleons 2 and 3). The resulting formulation opens a pathway toward a systematic investigation of the complex singularities in few-body scattering processes within the Faddeev framework.</p>