The complete symmetry group of the generalised hyperladder problem (Q1827115)

The authors determine the most general form of a system of \(n\) first-order ordinary differential equations under the symmetries given by \[ Y_m^\ell= \frac {x_m}{u^{1+a_m-a_\ell}} \left( \sum_{j=1}^n x_j \partial_{x_j}- u\partial_{x_\ell} \right). \tag{1} \] The authors show that the system must have the form \[ \dot x_i= \left[ x_i \sum_{j=1}^n (1+a_i-a_j) x_j\right] \varphi(t,u), \tag{2} \] where \(\varphi\) is an abitrary function of its arguments, which is somewhat more general than the homogeneous Lotka-Volterra system treated. The authors present some features of the algebraic structure of the symmetries given in (1) which have not been previously reported.