  Covariant affine integral quantization is studied and applied to the motion
of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space
is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer
operators built from weight functions on this phase space. To illustrate the
procedure, we examine two examples of weights. The first one corresponds to 2-D
coherent state families, while the second one corresponds to the affine
inversion in the punctured plane. The later yields the usual canonical
quantization and a quasi-probability distribution (2-D affine Wigner function)
which is real, marginal in both position and momentum.
