In this thesis we define the notion of an overshear on a Danielewskisurface. Next we show that the group generated by the overshears is dense in the component of the identity of the automorphism group. Moreover, we show that the overshear group has a structure of an amalgamated product, and as consequence of this the overshear group is a proper subgroup of the automorphism group. Finally we classify the R^n-actions, and therefore the one parameter subgroups, of the overshear group. We also show that any Lie subgroup of an amalgamated product can be conjugated to one of the factors of the amalgamated product.