Minimal Cubature rules and polynomial interpolation in two variables II

As a complement to \cite{X12}, minimal cubature rules of degree $4m+1$ for the weight functions $$ \mathcal{W}_{\alpha,\beta ,\pm \frac12}(x,y) = |x+y|^{2\alpha+1} |x-y|^{2\beta+1} ((1-x^2)(1-y^2))^{\pm \frac12} $$ on $[-1,1]^2$ are shown to exist and near minimal cubature rules of the same degree with one node more than minimal are constructed explicitly. The Lagrange interpolation polynomials on the nodes of the near minimal cubature rules are also studied.