Symplectic resolutions of the quotient of R2 by an infinite symplectic discrete group

We construct smooth symplectic resolutions of the quotient of R2 under some infinite discrete sub-group of GL2(R) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val symplectic resolution of C2/G, with G subset of SL2(C) a finite group. The first of these infinite groups is G=Z, identified to triangular matrices with spectrum {1}. Smooth functions on the quotient R2/G come with a natural Poisson bracket, and R2/G is for an arbitrary k >= 1 set-isomorphic to the real Du Val singular variety A2k={(x,y,z)is an element of R3,x2+y2=z2k}. We show that each one of the usual minimal resolutions of these Du Val varieties are symplectic resolutions of R2/G. The same holds for G '=Z Z/2Z (identified to triangular matrices with spectrum {+/- 1}), with the upper half of the Du Val singularity D2k+1 playing the role of A2k.