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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 thelevel of the index by the action of the finite difference operators on thecorresponding indices. There exist at least three different types of suchoperators 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 andderive explicit expression for an infinite tower of difference operatorsgeneralizing the van Diejen model. We check various properties of theseoperators originating from the geometry of compactifications. We also providean expression for the kernel function of both our $C_2$ operator and previouslyderived $A_2$ generalization of van Diejen model. Finally we also considercompactifications with $A_N$-type punctures and derive the full tower ofcommuting difference operators corresponding to this root system generalizingthe result of our previous paper.

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