About linear homogeneous hypersurfaces in $ \Bbb R^4 $

The article is related to the describing problem of affinely homogeneous hypersurfaces in the space $ \Bbb R^4 $ that have exactly $3$-dimensional affine symmetry algebras. For three types of solvable $3$-dimensional Lie algebras, their linearly homogeneous $3$-dimensional orbits in this space are studied, different from surfaces of the second order and cylindrical surfaces in $ \Bbb R^4 $ (which are of no interest in the problem under discussion).
The presence of two nontrivial commutation relations in each of the studied algebras leads to the essential difference between the situation with their orbits in $ \Bbb R^4 $ and the case of a $3$-dimensional Abelian algebra with a large family of affinely distinct (linearly homogeneous) orbits in the same space. It is proved that one of the studied types of Lie algebras does not admit nontrivial $4$-dimensional linear representations at all; a large number of $3$-dimensional orbits of representations of the other two types have rich symmetry algebras. At the same time for one of the three types of Lie algebras, a new family of linearly homogeneous orbits is obtained, which have precisely $3$-dimensional algebras of affine symmetries.