In this article the authors reveal a number of pedagogical conditions necessary for training the linguistically gifted school students in a foreign language in the system of additional foreign-language education providing fully specific content, the forms and methods of training and education of pupils aimed at the solution of the specific pedagogical objectives. Using the specific language material, the teacher of a foreign language (FL) has to try to create such pedagogical conditions in the system of additional foreign-language education that would teach the linguistically gifted school students to understand the outlook and psychology of the native-speakers of a FL which are generally expressed through artistic means having a set of uniform philosophical and psychological, mythopoetic sources in the ancient times which promotes the formation of the inner world of linguistically gifted school students in many respects.