We consider linear differential equations that depend on an additional parameter. The symmetry group of such an equation is called parameterized differential Galois group and it is a linear differential algebraic group, which means that it can be identified with a subgroup of GL_n defined by differential-algebraic equations. I will speak on the corresponding inverse problem over k((t))(x).

Using the field patching method developed by Harbater and Hartmann, it can be shown that some classes of linear differential algebraic groups occur as parameterized differential Galois groups. On the other hand, surprisingly many linear differential algebraic groups do not occur.