A birational transformation
\Phi : \mathbb {P}^n \dasharrow Z \subset \mathbb {P}^N, where
Z
⊂
P
N
Z \subset \mathbb {P}^N
is a nonsingular variety of Picard number 1, is called a special birational transformation of type
(
a
,
b
)
(a,b)
if
Φ
\Phi
is given by a linear system of degree
a
a
, its inverse
Φ
−
1
\Phi ^{-1}
is given by a linear system of degree
b
b
and the base locus
S
⊂
P
n
S \subset \mathbb {P}^n
of
Φ
\Phi
is irreducible and nonsingular. In this paper, we classify special birational transformations of type
(
2
,
1
)
(2,1)
. In addition to previous works by Alzati-Sierra and Russo on this topic, our proof employs natural
C
∗
\mathbb {C}^*
-actions on
Z
Z
in a crucial way. These
C
∗
\mathbb {C}^*
-actions also relate our result to the prolongation problem studied in our previous work.