  The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional
subspaces of a vector space V which are isotropic with respect to two general
skew-symmetric forms; it is a Fano variety which admits an action of a torus
with a finite number of fixed points. In this work we study its equivariant
cohomology when $k = 2$; the central result of the paper is an equivariant
Chevalley formula for the multiplication of the hyper-plane class by any
Schubert class. Moreover, we study in detail the case of I$_2$Gr$(2,
\mathbb{C}^6)$, which is a quasi-homogeneous variety, we analyze its
deformations and we give a presentation of its cohomology.
