  We consider supersymmetric surface defects in compactifications of the $6d$
minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann
surface. For the case of $N=1$ such defects are introduced into the
supersymmetric index computations by an action of the $BC_1\,(\sim A_1\sim
C_1)$ van Diejen model. We (re)derive this fact using three different field
theoretic descriptions of the four dimensional models. The three field
theoretic descriptions are naturally associated with algebras $A_{N=1}$,
$C_{N=1}$, and $(A_1)^{N=1}$. The indices of these $4d$ theories give rise to
three different Kernel functions for the $BC_1$ van Diejen model. We then
consider the generalizations with $N>1$. The operators introducing defects into
the index computations are certain $A_{N}$, $C_N$, and $(A_1)^{N}$
generalizations of the van Diejen model. The three different generalizations
are directly related to three different effective gauge theory descriptions one
can obtain by compactifying the minimal $(D_{N+3},D_{N+3})$ conformal matter
theories on a circle to five dimensions. We explicitly compute the operators
for the $A_N$ case, and derive various properties these operators have to
satisfy as a consequence of $4d$ dualities following from the geometric setup.
In some cases we are able to verify these properties which in turn serve as
checks of said dualities. As a by-product of our constructions we also discuss
a simple Lagrangian description of a theory corresponding to compactification
on a sphere with three maximal punctures of the minimal $(D_5,D_5)$ conformal
matter and as consequence give explicit Lagrangian constructions of
compactifications of this 6d SCFT on arbitrary Riemann surfaces.
