  With the aim of better understanding the class of 4D theories generated by
compactifications of 6D superconformal field theories (SCFTs), we study the
structure of N = 1 supersymmetric punctures for class S_Gamma theories, namely
the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes
probing a C^2 / Z_k singularity, the punctures are governed by a dynamical
system in which evolution in time corresponds to motion to a neighboring node
in an affine A-type quiver. Classification of punctures reduces to determining
consistent initial conditions which produce periodic orbits. The study of this
system is particularly tractable in the case of a single M5-brane. Even in this
"simple" case, the solutions exhibit a remarkable level of complexity: Only
specific rational values for the initial momenta lead to periodic orbits, and
small perturbations in these values lead to vastly different late time
behavior. Another difference from half BPS punctures of class S theories
includes the appearance of a continuous complex "zero mode" modulus in some
puncture solutions. The construction of punctures with higher order poles
involves a related set of recursion relations. The resulting structures also
generalize to systems with multiple M5-branes as well as probes of D- and
E-type orbifold singularities.
