We construct smooth symplectic resolutions of the quotient of R^2 under some infinite discrete sub-group of GL_2(R) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val symplectic resolution of C^2/G, with G \subset SL_2(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 R^2/G come with a natural Poisson bracket, and R^2/Gis for an arbitrary k \geq 1 set-isomorphic to the real Du Val singular variety A_{ 2k} = { (x,y,z) \in R^3 , x^2 +y^2= z^{ 2k}}. We show that each one of the usual minimal resolutions of these Du Val varieties are symplectic resolutions of R^2/G. The same holds for G'=Z \rtimes Z/2Z (identified to triangular matrices with spectrum { \pm 1}), with the upper half of D_{ 2k+1} playing the role of A_{ 2k}.
Categories: math.DG