A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C is introduced. It is proved that BA function exists only for very special configurations (locus… (More)

The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the… (More)

A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding… (More)

We consider a complex version of the ∨-systems, which appeared in the theory of the WDVV equation. We show that the class of these systems is closed under the natural operations of restriction and… (More)

Initial value problems for the integrable discrete equations on quadgraphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete… (More)

We present a new family of the locus configurations which is not related to ∨-systems thus giving the answer to one of the questions raised in [1] about the relation between the generalised quantum… (More)

It is proved that if the Schrödinger equation Lψ = λψ of CalogeroMoser-Sutherland type with L = −∆ + ∑ α∈A+ mα(mα + 1)(α, α) sin(α, x) has a solution of the product form ψ0 = ∏ α∈A+ sin −mα(α, x),… (More)

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and… (More)

A generalization of the theory of algebro-geometric Poisson brackets on the space of finite-gap Schrödinger operators, developped by S. P. Novikov and A. P. Veselov, to the case of periodic… (More)

A generalisation of the Stieltjes relations for the Painlevé-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a… (More)