Ga-actions and completions

Let δ be a locally nilpotent derivation on an affine domain B defined over the complex field C and let A=Kerδ. Let M be a maximal ideal of B and let m=A∩M. Then δ extends to C-derivations δM and on the local ring BM and its M-adic completion . We shall show that is not necessarily equal to the m-adic completion , though KerδM=Am provided B is factorial. This gives a negative answer to a problem raised in [M. Miyanishi, Problems in Mathematisches Forschungsinstitut Oberwolfach Report No. 01/2007 (p. 70) on the workshop “Affine Algebraic Geometry”, 2007. [5]]. As a related result, we also give an example of a Ga-equivariant, nonfinite étale endomorphism φ of a smooth affine surface Y with a Ga-action for which the induced endomorphism ψ on the algebraic quotient X=Y//Ga is ramified.