We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system. 1 2 n(n + 1) = 6 Killing vectors (when n = 3).The conformal algebra of this 3D metric has dimension 1 2 (n + 1)(n + 2) = 10 (when n = 3). A convenient

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The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials.

We perform preliminary group classification of a class of fourth-order evolution equations in one spatial variable. Following the approach developed in [1] we construct all inequivalent partial differential equations belonging to the class in question which admit semi-simple Lie groups. In addition, we describe all fourth-order evolution equations from the class under consideration which are invariant under solvable Lie groups of dimension n <= 4. We have constructed all Galilei-invariant equations belonging to the class of evolution differential equations under study. The list of so obtained invariant equations contains both the well-known fourth-order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics, chemistry and biology.

ABSTRACT. We consider the weakly coupled system of reaction-diffusion equations ut = Au + \x\ ai vP, vt=Av+ |a:r 2 it 9 ,where x E R^ (AT > 1), t > 0, p, q > 1 with pq > 1 and 0 < ai < N(jp -1), 0 < (72 < N(q -1). Put

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Abstract. Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the "kinetic energy", related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.

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