  The metrizability problem for a symmetric affine connection on a manifold,
invariant with respect to a group of diffeomorphisms G, is considered. We say
that the connection is G-metrizable, if it is expressible as the Levi-Civita
connection of a G-invariant metric field. In this paper we analyze the
G-metrizability equations for the rotation group G = SO(3), acting canonically
on three- and four-dimensional Euclidean spaces. We show that the property of
the connection to be SO(3)-invariant allows us to find complete explicit
description of all solutions of the SO(3)-metrizability equations.
