Exact solutions to a system of ODEs arising in classification of diagonalizable hydrodynamic chains

We obtain exact solutions of a system of two third--order ordinary differential equations arising in the classification of diagonalizable hydrodynamic chains. We show that the system admits a four--dimensional solvable Lie algebra of point symmetries. Although this algebra is not large enough for the complete integrability by quadratures, it enables a reduction to a two--dimensional first--order system. A further symmetry reduction leads to an Abel equation whose implicit solutions define a family of algebraic curves. By analysing their singular cases and deriving rational parametrizations, we obtain the solutions to the reduced system in parametric form. The solutions to the original third--order system are then reconstructed by four successive quadratures using Lie--Bianchi's theorem. The resulting families contain a previously known specific solution as a particular case and provide new diagonalizable hydrodynamic chains.