  The orbit-sum method is an algebraic version of the reflection-principle that
was introduced by Bousquet-M\'{e}lou and Mishna to solve functional equations
that arise in the enumeration of lattice walks with small steps restricted to
$\mathbb{N}^2$. Its extension to walks with large steps was started by Bostan,
Bousquet-M\'{e}lou and Melczer. We continue it here, making use of the
primitive element theorem, Gr\"{o}bner bases and the shape lemma, and the
Newton-Puiseux algorithm.
