Abstract. We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang-Mills equations in four dimensions. In particular the Jackiw-Pi vortex and the Ambjørn-Olesen vortex correspond to the gauge group SU (1, 1), and respectively the Euclidean or the SU (2) symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang-Mills theory in 4+1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of noncompact gauge groups.

We solve the metrisability problem for the six Painlevé equations, and more generally for all 2nd order ODEs with Painlevé property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.A problem of characterising metrisable ODEs by differential invariants was posed by Roger Liouville [21], who has reduced it to an overdetermined system of linear PDEs (see Theorem 2.1 in the next section). The complete solution was provided relatively recently [1], where it was shown that an ODE is metrisable if and only if three point invariants of differential orders five and six vanish, and certain genericity assumptions hold.A different approach was developed by Painlevé, Kowalevskaya and Gambier who studied 2nd order ODEs in the complex domain [24,16].Definition 1.2. The ODE y ′′ = R(x, y, y ′ ), where R is a rational function of y and y ′ has the Painlevé property (PP) if its movable singularities (i.e. singularities whose locations depend on the initial conditions) are poles.The solutions of equations with Painlevé property are single-valued thus giving rise to proper functions on C. There exists fifty canonical types of second order ODEs with PP

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We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin-Novikov type for a given one-dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from two-dimensional Frobenius manifolds.

We propose a modified version of the Ginzburg-Landau energy functional admitting static solitons and determine all the Painlevé-integrable cases of its Bogomolny equations of a given class of models. Explicit solutions are determined in terms of the third Painlevé transcendents, allowing us to calculate physical quantities such as the vortex number and the vortex strength. These solutions can be interpreted as the usual Abelian-Higgs vortices on surfaces of non-constant curvature with conical singularity.