  A class of the $D=4$ gravity models describing a coupled system of $n$
Abelian vector fields and the symmetric $n \times n$ matrix generalizations of
the dilaton and Kalb-Ramond fields is considered. It is shown that the
Pecci-Quinn axion matrix can be entered and the resulting equations of motion
possess the $Sp(2n, R)$ symmetry in four dimensions. The stationary case is
studied. It is established that the theory allows a $\sigma$-model
representation with a target space which is invariant under the $Sp[2(n+1), R]$
group of isometry transformations. The chiral matrix of the coset $Sp[2(n+1),
R]/U(n+1)$ is constructed. A K\"ahler formalism based on the use of the Ernst
$(n+1) \times (n+1)$ complex symmetric matrix is developed. The stationary
axisymmetric case is considered. The Belinsky-Zakharov chiral matrix depending
on the original field variables is obtained. The Kramer-Neugebauer
transformation, which algebraically maps the original variables into the target
space ones, is presented.
