We study the geometry of multidimensional scalar 2 nd order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M (1) over a (2n + 1)-dimensional contact manifold (M, C). We develop the theory of characteristics of the equation E in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of E. After specifying the results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in [11], i.e. MAEs of the formWe show that any MAE of the aforementioned class is associated with an n-dimensional subdistribution D of the contact distribution C, and viceversa. We characterize this Goursat-type equations together with its intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method of solutions of a Cauchy problem, provided the existence of a suitable number of intermediate integrals.

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

We study the geometry of differential equations determined uniquely by their point symmetries, that we call Lie remarkable. We determine necessary and sufficient conditions for a differential equation to be Lie remarkable. Furthermore, we see how, in some cases, Lie remarkability is related to the existence of invariant solutions. We apply our results to minimal submanifold equations and to Monge-Ampère equations in two independent variables of various orders.

For each simple Lie algebra g (excluding, for trivial reasons, type C) we find the lowest possible degree of an invariant 2 nd order PDE over the adjoint variety in Pg, a homogeneous contact manifold. Here a PDE F (x i , u, u i , u ij ) = 0 has degree ≤ d if F is a polynomial of degree ≤ d in the minors of (u ij ), with coefficients functions of the contact coordinates x i , u, u i (e.g., Monge-Ampère equations have degree 1). For g of type A or G 2 we show that this gives all invariant 2 nd order PDEs. For g of type B and D we provide an explicit formula for the lowest-degree invariant 2 nd order PDEs. For g of type E and F 4 we prove uniqueness of the lowest-degree invariant 2 nd order PDE; we also conjecture that uniqueness holds in type D.

We give a complete list of mutually non-diffeomorphic normal forms for the two-dimensional metrics that admit one essential (i.e., non-homothetic) projective vector field. This revises some results in [25] and extends the results of [10,25], solving a problem posed by Sophus Lie in 1882 [22]. MSC 2010 classes: 53A20, 53A55, 53B10 Keywords: projective connections; projective symmetries; projectively equivalent metrics Problem 1 (Lie, 1882). Determine the metrics that describe surfaces whose geodesic curves admit an infinitesimal transformation 1 , i.e. metrics whose projective algebra has dim(p(g)) ≥ 1.Fubini referred to this problem as the "Lie problem", see [1,16]. An overview of the history of the problem can be found, for instance, in [1,2]. Important works in the field, with respect to the considerations in the present paper, are for instance [14,23,10,25,9].Problem 1 is also referred to as Lie's First Problem, in contrast to the narrower problem when the inequality dim(p(g)) > 1 holds strictly. The latter problem is then referred to as Lie's Second Problem. A solution to Lie's (Second) Problem was first claimed in [1,2], but the proof contained a gap. A correct solution is given in [10], where mutually non-diffeomorphic normal forms of 2-dimensional metrics g with dim(p(g)) ≥ 2 were found around generic points (i.e., the orbit of the projective algebra is of constant non-zero dimension in a neighborhood of these points). Metrics admitting exactly one, essential projective vector field have been treated in [25], around generic points, providing an explicit list of projective classes. However, this list is not a list of mutually non-diffeomorphic normal forms, i.e. non-isometric metrics; actually, it is not even sharp as a list of projective classes under projective transformations. Moreover, reference [25] contains some gaps and some supplementary results are incomplete, see Section 2.1 where these issues are discussed in detail, along with a brief outline of the state of art. The main outcomes of the present paper are discussed in Section 2.2: Theorems 2, 3 and 4 together with Proposition 5, constitute a classification in terms of normal forms up to isometries of metrics that admit exactly one, essential projective vector field. Note that Theorems 2 and 3 constitute corrections of two results found in [25], namely Theorems 2 and 3 of this reference. As a by-product of the proof of Theorem 4, we also obtain a classification of all projective classes that cover metrics with exactly one, projective vector field that is essential (as stated above, [25] provides only a description of such classes, not a classification, see Section 2.1 for more details).For the formulation of the main results, we need the following proposition.1 German original [22]: "Es wird verlangt, die Form des Bogenelementes einer jeden Fläche zu bestimmen, deren geodätische Curven eine infinitesimale Transformation gestatten." 2 In the current context, we use Proposition 1 to describe a pair of projectively equivalent metrics that serve as "...