  We classify entire positive singular solutions to a family of critical sixth
order equations in the punctured space with a non-removable singularity at the
origin. More precisely, we show that when the origin is a non-removable
singularity, solutions are given by a singular radial factor times a periodic
solution to a sixth order IVP with constant coefficients. On the technical
level, we combine integral sliding methods and qualitative analysis of ODEs,
based on a conservation of energy result, to perform a topological
two-parameter shooting technique. We first use the integral representation of
our equation to run a moving spheres technique, which proves that solutions are
radially symmetric with respect to the origin. Thus, in Emden--Fowler
coordinates, we can reduce our problem to the study of an sixth order
autonomous ODE with constant coefficients. The main heuristics behind our
arguments is that since all the indicial roots of the ODE operator are
positive, it can be decomposed into the composition of three second order
operators satisfying a comparison principle. This allows us to define a
Hamiltonian energy which is conserved along solutions, from which we extract
their qualitative properties, such as uniqueness, boundedness, asymptotic
behavior, and classification.
