The class of distributions on Rᵐ, 1 ≤ m < ∞ which are the m-dimensional marginal distributions of orthogonally invariant distributions on Rm + n is characterized. This result is then used to provide a partial answer to the following question: given a symmetric distribution on R¹ and an integer n ≥ 2, under what conditions will there exist a random vector X ∈ Rⁿ such that $a'X$ has the given distribution (up to a positive scale factor) for all a ≠ 0, a ∈ Rᵐ.
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Morris L. Eaton (1981) studied this question.