Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity, symmetry and integrability properties. We examine a variation of this sequence and find that there are some remarkable changes in properties consequential upon this variation.

We compute invariants for the two-variable M\"obius transformation. In particular we are interested in partial differential equations in two dependent and two independent variables that are kept invariant under this transformation.

We introduce a new type of recursion operator suitable to generate a class of nonlocal symmetries for those second-order evolution equations in 1+1 dimension which allow the complete integration of their time-independent versions. We show that this class of evolution equations is C-integrable (linearizable by a point transformation). We also discuss some applications.

We present a differential sequence based upon the Calogero-Degasperis-Ibragimov-Shabat Equation and determine first integrals and the general solution. Under suitable transformations eachmember of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form.

Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the ‘exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.

We define a proper differential sequence of ordinary differential equations and introduce a method for deriving an alternate sequence of integrals for such a sequence. We describe some general properties, illustrated by several examples.

We discuss the construction of reciprocal Bäcklund transformations for evolution equations using integrating factors of zeroth and higher orders with their corresponding conservation laws. As an example, we consider the Harry Dym equation and the Schwarzian KdV equation.

Certain nonlinear evolution PDEs in 1+1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T=2π/ω (or perhaps T˜=pT with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited.

We consider u t = u α u xxx + n ( u ) u x u xx + m ( u ) u 3 x + r ( u ) u xx + p ( u ) u 2 x + q ( u ) u x + s ( u ) with α= 0 and α= 3 , for those functional forms of m , n , p , q , r , s for which the equation is integrable in the sense of an infinite number of Lie‐Bäcklund symmetries. Recursion operators which are x ‐ and t ‐independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x ‐generalized hodograph transformations are applied to the obtained equations.

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form x((n)) = h (x, x((n-1))) x over dot are considered. The relation of the proposed method to local and nonlocal symmetries are discussed. (C) 2003 Published by Elsevier Inc.

We calculate in detail the conditions which allow the most general third-order ordinary differential equation to be linearised in X′′′(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t) dt.

In our article "A tree of linearisable second-order evolution equations by generalised hodograph transformations" [J. Nonlin. Math. Phys. {\bf 8} (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution equations in (1+1) dimensions. Expanding this result we report here the complete set of recursion operators for this tree and present several linearisable (C-integrable) hierarchies in (1+1) dimensions.

We propose several approaches for solving two discrete-velocity Boltzmann equations using the rescaling ansatz and the truncated Painlevé expansions. We use solutions of the two- and three-dimensional Bateman equations for the singularity manifold conditions to reduce the problem to Riccati equations. Both equations fail the Painlevé test.

Marianna EULER, Norbert EULER and Niclas PETERSSON Department of Mathematics, Lule̊a University of Technology, SE-971 87 Lule̊a, Sweden E-mails: Norbert@sm.luth.se, Marianna@sm.luth.se Submitted for publication in the Proceedings of the Öresund Symposium on Partial Differential Equations; May 23-25, 2002 Abstract In our article [5], “A tree of linearisable second-order evolution equations by generalised hodograph transformations [J. Nonlin. Math. Phys. 8 (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution equations in (1+1) dimensions. Expanding this result we report here the complete set of recursion operators for this tree and present several linearisable (C-integrable) hierarchies in (1+1) dimensions.

We make use of the Painleve expansion and the general implicit solution of the multidimensional Bateman equation to construct explicit solutions of two discrete velocity Boltzmann equations in (1 + 1) and (1 + 2) dimensions, respectively.

We introduce nonlocal auto-hodograph transformations for a hierarchy of nonlinear evolution equations. This is accomplished by composing nonlocal transformations (one of which is a hodograph transformation) which linearize the given equations. This enables one to construct sequences of exact solutions for any equation belonging to the hierarchy.

We reduce the nonlinear wave equation square (n)u = alphaF[exp(betau)] to ordinary differential equtions and construct exact solutions, by the use of a compatible d'Alembert-Hamilton system. The solutions of these ordinary differential equations, together with the solutions of the corresponding d'Alembert-Hamilton equations, provide a rich class of exact solutions of the multidimensional wave equations. The wave equations are studied in n-dimensional Minkowski space.

The general d'Alembert equation □u + f (x0, x1, u) = 0 is considered, where □ is the two-dimensional d'Alembert operator. We classify the equation for functions f by which it admits several Lie symmetry algebras, which include the Lorentz symmetry generator. The corresponding symmetry reductions are listed.

The Madelung representation = uexp(iv) is considered for the d’Alembert equation 2n F(| |) = 0 to develop a technique for finding exact solutions. We classify the nonlinear function F for which the amplitude and phase of the d’Alembert equation are related to the solutions of the compatible d’Alembert‐Hamiltonian system. The equations are studied in n-dimensional Minkowski space. We consider the following general nonlinear d’Alembert equation