Abstract: We discuss a method to construct Dirac-harmonic maps developed by Jürgen Jost, Xiaohuan Mo and Miaomiao Zhu in Some explicit constructions of Dirac-harmonic maps, J. Geom. Phys. 59 (2009), no. 11, 1512-1527. The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in Some explicit constructions of Dirac-harmonic maps, J. Geom. Phys. 59 (2009), no. 11, 1512-1527, were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions. In several cases we show the existence of solutions.