This paper presents a method for constructing normal solutions of inhomogeneous systems of second-order partial differential equations of rank kkk. Special attention is given to the case when the subrank of the system is equal to one, which simplifies the procedure for determining the undetermined parameters. Analytical methods are used in the study, including the Frobenius-Latysheva transformation and the representation of solutions in the form of generalized power series. Necessary conditions for the existence of normal solutions are established, and an analysis of auxiliary systems is conducted. Using the Frobenius-Latysheva method, requirements for recurrent systems that determine the unknown coefficients of normal solutions are formulated. It is shown that when these requirements are met, the necessary conditions for the existence of a solution also become sufficient. The obtained results can be applied to problems in mathematical physics, engineering, and other fields of science.