Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_∗=R^2{0, for which the phase space is R^2_∗=R^2{0X 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.