# Existence of a global analytic solution to a linear first order PDE

Let $$B=\lbrace \|z\|<1\rbrace$$ be a unit ball in $$\mathbb{C}^n, n\geq 2.$$ Let $$f_1,\cdots, f_n, f$$ be holomorphic functions on $$B.$$ Now, consider the following first order, linear PDE: $$f_1\partial_{z_1}(y)+\cdots +f_n\partial_{z_n}(y)+fy=0.$$ Here is my question. Is there some non-degeneracy condition of functions $$f_1,\cdots, f_n$$ that implies existence of a nontrivial solution $$y$$ which is holomorphic on $$B?$$ From what I understand about the Cauchy-Kovalevskaya theorem, if $$f_1,\cdots, f_n$$ have no common zero, then for any $$w\in B$$ there exists a nontrivial holomorphic solution (in fact infinitely many) defined near $$w.$$ But, I don't see how to get a global solution from it.Thank you very much in advance.

If the holomorphic vector field $$X=\sum_i f_i\partial_{z_i}$$ has no zeroes and there is a complex hypersurface nowhere tangent to it, and each point lies on a unique trajectory of that vector field through that hypersurface, then the differential equation becomes a linear ordinary differential equation for a holomorphic parameterization of each trajectory, so has global solvability. So you need to find your nowhere tangent hypersurface, and prove, for each point, both existence and uniqueness of a trajectory through that hypersurface and through that point.