Successive Lie Reduction and Integration of a Ferapontov-Type First-Order System of Ordinary Differential Equations

This article develops a symmetry-based integration programme for a first-order system with two dependent variables obtained in the classification of diagonalisable hydrodynamic chains. Represented in the three-dimensional space with coordinates x,y,z, the system admits a non-trivial Lie point symmetry which reduces it to an Abel differential equation of the second kind for βr, together with an auxiliary quadrature for αr. The central observation pursued here is that the reduced differential equation itself admits a nontangent Lie point symmetry. This makes it possible to interpret the integration process as a genuine successive reduction procedure: first by the Lie point symmetry of the two-dimensional first-order system and then by a symmetry of the scalar reduced equation. On regular local branches, the resulting solutions are described by a rational first integral in the r,β-plane, by explicit parametrisations of the corresponding generic algebraic curves, and by an inverse transformation back to x,y,z. The emphasis is on the successive-reduction mechanism underlying integration of the two-dimensional first-order system.