In this paper, we provide a deformation-curl-Poisson decomposition for the three dimensional steady Euler-Poisson system. By using the Bernoulli's law, we rewrite the density equation as a Frobenius inner product of a symmetric matrix and the deformation matrix with an additional term reflecting the electrostatic effect. The vorticity will be solved by a transport equation with another two algebraic equations, hence one obtains a nonlinear system consisting of the deformation equation, the curl system and the Poisson equation, which is elliptic in the sense of Agmon, Dougalis and Nirenberg. As an application, we establish the structural stability of 1-D subsonic background solutions with multidimensional perturbations on the entrance and exit of a rectangular cylinder. The key ingredient of the analysis lies in some a priori estimates for the deformation-curl-Poisson system with tangential or normal boundary conditions for the velocity.