  In the present paper, we discuss the singular minimal surfaces in a Euclidean
3-space R^{3} which are minimal. In fact, such a surface is nothing but a
plane, a trivial outcome. However, a non-trivial outcome is obtained when we
modify the usual condition of singular minimality by using a special
semi-symmetric metric connection instead of the Levi-Civita connection on
R^{3}. With this new connection, we prove that, besides planes, the singular
minimal surfaces which are minimal are the generalized cylinders, providing
their explicit equations. A trivial outcome is observed when we use a special
semi-symmetric non-metric connection. Furthermore, our discussion is adapted to
the Lorentz-Minkowski 3-space.
