$C_2$ generalization of the van Diejen model from the minimal $(D_5,D_5)$ conformal matter

We study superconformal indices of $4d$ compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with $A_N, C_N$ and $\left(A_1\right)^N$ global symmetries. We mainly concentrate on $C_2$ case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our $C_2$ operator and previously derived $A_2$ generalization of van Diejen model. Finally we also consider compactifications with $A_N$-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.