On the structure of solutions of a degenerate elliptic system (Q1897876)

The author considers a system of the form \[ x \left( I_1 {\partial u \over \partial x} + I_2 {\partial u \over \partial y} + I_3 {\partial u \over \partial z} \right) + A(x,y,z) u = 0, \quad x,y,z \in \mathbb{C}, \tag{1} \] where \(I_1, I_2, I_3\) are constant coefficient matrices and \(A\) is a holomorphic matrix function on the polycylinder \(P = \{(x,y,z) \mid |x |< r_1\), \(|y - y_0 |< r_2\), \(|z - z_0 |< r_3\}\) representable by the series \(A(x,y,z) = \sum^\infty_{k = 0} A^{(k)} (y,z) x^k\). It is proved that (1) has a solution represented by a series of the form \[ u(x,y,z) = x^\rho \sum^\infty_{k = 0} \sum^k_{l = 0} x^{k - l} v_l^{(k)} (y,z) (x \log x)^l. \]