  In our previous works (2012, 2013), we provided a finite list of properties
characterizing all potential types of quadratic birational transformations of a
projective space into a factorial variety, whose base locus is smooth and
irreducible. However, some existence problems remained open. Among them one had
to prove that the image of a given transformation was factorial, but in two
particular situations, even the mere existence of the transformation was left
as an open problem. In this paper, we use computer algebra methods to construct
explicitly four examples of such transformations; two of them were among those
for which it was not known that the image was factorial, and the other two show
the existence of the two above referred transformations.
