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[en] Starting with the first-order formulation of the massless superparticle model on the superbackground and presenting the momentum components tangent to and subspaces as bilinear combinations of the constrained -Majorana spinors allows to bring the superparticle's Lagrangian to the form quadratic in supertwistors. The supertwistors are assembled into a pair of doublets, one of which has even and odd components, while the other has odd and even components. They are subject to the first-class constraints that generate the gauge algebra. This justifies previously proposed group-theoretic definition of the supertwistors and allows to derive the incidence relations with the supercoordinates of the superspace. Whenever superparticle moves within the subspace of the space-time, twistor formulation of its Lagrangian involves just one doublet of supertwistors with even and odd components. If in addition particle's 5-momentum is null, four first-class constraints which are the generators single out upon quantization the states of gauged supergravity multiplet in the superambitwistor formulation.