Study the behaviour and the evolution of the cosmological field equations in an homogeneous and anisotropic spacetime with two scalar fields coupled in the kinetic term. Specifically, the kinetic energy for the scalar field Lagrangian is that of the Chiral model and defines a two-dimensional maximally symmetric space with negative curvature. For the background space we assume the locally rotational spacetime which describes the Bianchi I, the Bianchi III and the Kantowski-Sachs anisotropic spaces. We work on the $H$% -normalization and we investigate the stationary points and their stability. For the exponential potential we find a new exact solution which describes an anisotropic inflationary solution. The anisotropic inflation is always unstable, while future attractors are the scaling inflationary solution or the hyperbolic inflation. For scalar field potential different from the exponential, the de Sitter universe exists.
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