  The formal derivatives of the Yang-Baxter equation with respect to its
spectral parameters, evaluated at some fixed point of these parameters, provide
us with two systems of differential equations. The derivatives of the $R$
matrix elements, however, can be regarded as independent variables and
eliminated from the systems, after which two systems of polynomial equations
are obtained in place. In general, these polynomial systems have a non-zero
Hilbert dimension, which means that not all elements of the R matrix can be
fixed through them. Nevertheless, the remaining unknowns can be found by
solving a few number of simple differential equations that arise as consistency
conditions of the method. The branches of the solutions can also be easily
analyzed by this method, which ensures the uniqueness and generality of the
solutions. In this work we considered the Yang-Baxter equation for the
$(n+1)(2n+1)$-vertex models with a generalization based on the $A_n$ symmetry.
This differential approach allowed us to solve the Yang-Baxter equation in a
systematic way.
