Differential equations uniquely determined by algebras of point symmetries

In a recent paper \cite{MannoOliveriVitolo06}, within the framework of the inverse Lie problem, the definitions of strongly as well as weakly Lie remarkable equations have been introduced. Strongly Lie remarkable equations are uniquely determined by their Lie point symmetries, whereas weakly Lie remarkable equations are equations which do not intersect other equations admitting the same symmetries. In this paper we start from some relevant algebras of vector fields on $\R^k$ (such as the isometric, affine, projective or conformal algebra), and characterize strongly Lie remarkable equations admitted by the considered Lie algebras.