  The paper contributes to the important and urgent problem to extend the
physical theory of space-time in a Finsler-type way under the assumption that
the isotropy of space is violated by a single geometrically distinguished
spatial direction which destroys the pseudo-Euclidean geometric nature of the
relativistic metric and space. It proves possible to retain the fundamental
geometrical property that the indicatrix should be of the constant curvature.
Similar property appears to hold in the three-dimensional section space. The
last property was the characteristic of three-dimensional positive-definite
Finsleroid space proposed and developed in the previous work, so that the
present paper lifts that space to the four-dimensional relativistic level. The
respective pseudo-Finsleroid metric function is indicated. Numerous significant
tensorial and geometrical consequences have been elucidated.
  \ses {\bf Keywords:} Finsler metrics, relativistic spaces.
