An unsolvable Cousin problem

We construct a Laurent series $\sum\nolimits_{n = - \infty }^\infty {{A_n}{z^n}}$, convergent in an annulus $\left\{ {\rho < \left\vert z \right\vert < 1/\rho } \right\}$ for some $\rho < 1$, which satisfies an algebraic differential equation (ADE) there, but such that neither of the series $\sum\nolimits_{n = 0}^\infty {{A_n}{z^n}}$ and $\sum\nolimits_{n = - \infty }^{ - 1} {{A_n}{z^n}}$ satisfies any ADE.