. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of "totally singular points", and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations. y Supported in part by NSF Grant DMS 98...

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lamé potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic. 1

We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for third-order evolution equations that admit reduction to Cauchy problems for systems of two ordinary di#erential equations. These reductions are shown to correspond to higher conditional symmetries admitted by the corresponding nonlinear evolution equations.

Ordinary dierential operators possessing invariant subspaces spanned by the functions x , i = 0; 1; : : : ; n 1, are considered. A complete description of operators of the submaximal order n 2 is obtained. The dimension C 2n 1 of the linear space of such operators is conjectured to be the upper bound for the operators possessing arbitrary n-dimensional invariant subspace. A general representation for translation-invariant operators is found.

Abstract. In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables. AMS subject classification: 47F5, 35K55, 81R15 1.

We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order n that admit an N-shock type solution with N ≤ n + 1. To this end we develop a refinement of the technique from our earlier work [12] where we completely characterized all (1+1)-dimensional evolution systems ut = F(x,t,u,∂u/∂x,..., ∂ n u/∂x n) that are conditionally invariant under a given generalized (Lie–Bäcklund) vector field Q(x,t,u,∂u/∂x,..., ∂ k u/∂x k)∂/∂u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.

Geometric interplay between function subspaces and their rings of differential operators