  A $\mathbb{C}^{*}$-action on a projective variety $X$ is said to be of Euler
type at a nonsingular fixed point $x$ if the isotropy action of
$\mathbb{C}^{*}$ on $T_{x}X$ is by scalar multiplication. In this paper, it's
proven that a smooth projective variety of Picard number one $X$ is isomorphic
to an irreducible Hermitian symmetric space of tube type if and only if for a
general pair of points $x,y$ on $X$, there exists a $\mathbb{C}^{*}$-action on
$X$ which is of Euler type at $x$ and its inverse action is of Euler type at
$y$.
