Let f : M → R^p be a special generic map of a closed n-dimensional manifold M with n ≥ p ≥ 1. We study the condition for f to be factorized as f = π ο η for an immersion (or an embedding) η : M → Rⁿ⁺¹ and an orthogonal projection π : Rⁿ⁺¹ → R^p. For various dimension pairs (n,p) we give answers to such a lifting problem. In particular, for the cases where p is equal to 1 and 2 we obtain complete results.