Explicit Integrability of the Generalised Ladder Problem

Abstract

We provide the explicit solution of the n-dimensional generalised ladder system, that is the homogeneous quadratic system of first-order differential equations of the form \\dotx_i=x_i∑_j=1ⁿa_ijx_j, i=1,n, where (a_ij)=(1+a_i−a_j), i,j=1,n introduced by Imai and Hirata [J. Phys. Soc. Jpn. 72 (2003) 973]. These systems are characterised by the n²−1 symmetries Y_m^l=x_mu^{a_l−a_m−1}(∑_j=1ⁿx_j∂_xj−u∂_xl), but are not the most general systems invariant under these symmetries. The more general systems are called hyperladder systems and we discuss their integrability.