  By Heyde's theorem, the class of Gaussian distributions on the real line is
characterized by the symmetry of the conditional distribution of one linear
form of independent random variables given another. We prove an analogue of
this theorem for two independent random vectors taking values in the space
$\mathbb{R}^n$. The obtained class of distributions consists of convolutions of
Gaussian distributions and a distribution supported in a subspace, which is
determined by coefficients of the linear forms.
