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Finslerian extension of the theory of relativity implies that space-time can be not only in an amorphous state which is described by Riemann geometry but also in ordered, i.e. crystalline states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics. It is shown that there are only two types of curved Finslerian spaces endowed with local relativistic symmetry. However the metric of only one of them satisfies the correspondence principle with Riemannian metric of the general theory of relativity and therefore underlies viable Finslerian extension of the GR. Since the existing purely geometric approaches to a Finslerian generalization of Einstein's equations do not allow one to obtain such generalized equations which would provide a local relativistic symmetry of their solutions, special attention is paid to the property of the specific invariance of viable Finslerian metric under local conformal transformations of those fields on which it explicitly depends. It is this property that makes it possible to use the well-known methods of conventional field theory and thereby to circumvent the above-mentioned difficulties arising within the framework of purely geometric approaches to a Finslerian generalization of Einstein's equations.