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A Hauser-Est-type extended hyperbolic complex linear system given in our previous paper[Gao Y J 2004 Chin.Phys.13 602]is slightly modified and used to develop a new inverse scattering method for the stationary axisymmetric Einstein-Maxwell theory with multiple Abelian gauge fields.The reduction procedures in this inverse scattering method are found to be fairly simple,which makes the inverse scattering method be fine and effective in practical application.As an example,a concrete family of soliton solutions for the considered theory is obtained.