Gap between nonradial and radial cubic Hsiang solutions

Ascertain the extent of the difference between homogeneous cubic polynomial solutions u: R^n -> R of the nonradial Hsiang equation M(u) = -3 b(x,x) u, where b is an arbitrary quadratic form, and solutions of the radial Hsiang equation M(u) = -θ h(x,x) u, where h is the Euclidean quadratic form on R^n; in particular, determine whether there exist irreducible solutions of the nonradial Hsiang equation that are not also radial eigencubics.

Background

The paper studies cubic homogeneous polynomial solutions of the Hsiang equations arising from the minimal cone condition M(u) = Q u, where for cubic u the right-hand side Q must be a quadratic form. Two variants are considered: the nonradial Hsiang equation M(u) = -3 b(x,x) u with arbitrary quadratic form b, and the radial Hsiang equation M(u) = -θ h(x,x) u with h the Euclidean quadratic form.

It is known that products of three linear forms always solve the nonradial equation and only in special cases solve the radial one, showing that nonradial solutions exist beyond the radial class. However, every currently known irreducible (i.e., non-factorable) solution of the nonradial equation is also a radial eigencubic, suggesting that the gap between the two classes might be small. The authors explicitly note that the size of this gap is unclear.