  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.
