  We discuss a method to construct Dirac-harmonic maps developed by J.~Jost,
X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions
of Dirac-harmonic maps}, J. Geom. Phys. \textbf{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 J.~Jost, X.~Mo, M.~Zhu, \emph{Some
explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{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.
