We study a bosonic four-dimensional effective action corresponding to the heterotic string compactified on a six-torus (dilaton-axion gravity with one vector field) on a curved space-time manifold possessing a timelike Killing vector field. Previously the existence of the $\mathrm{SO}(2, 3)\sim \mathrm{Sp}(4, R)$ global symmetry ($U$ duality) as well as the symmetric space property of the corresponding $\sigma$ model have been established following the Neugebauer-Kramer approach. Here we present an explicit form of the $\mathrm{Sp}(4, R)$ generators in terms of coset variables and construct a representation of the coset in terms of the physical target space coordinates. A complex symmetric 2 × 2 matrix $Z$ ("matrix dilaton-axion") is then introduced for which $U$ duality takes the matrix-valued $\mathrm{SL}(2, R)$ form. In terms of this matrix the theory is further presented as a Kähler $\sigma$ model. This leads to a more economic 2 × 2 formulation suggesting some new solution-generating techniques. A new solution (corresponding to constant $\mathrm{Im}Z$) is obtained which describes the system of point massless magnetic monopoles endowed with axion charges. In such a system mutual magnetic repulsion is exactly balanced by axion attraction so that the resulting space-time is locally flat but possesses multiple Taub-NUT singularities.

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