On the Equation ∇u = uV

The PDE ∇u = uV can be treated as its ODE equivalent u = au if the vector field V satisfies the integrability condition curl V = 0. In this case there exist solutions v of the equation ∇v = V and solutions of the original equation have the form u = cev (c ∈ ). However, in devising meaningful solution concepts, the equation displays some surprising features related to local and global integrability of its solutions. In the course of the investigation technical tools like a lifting theorem with respect to the exponential map and Poincar´e type theorems are proved for distributions and functions of Beppo Levi type.