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In this paper we prove a version of Lie-Bäcklund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the plane treated by E. Cartan in 1910. The approach is based on the geometric theory of PDEs and Tanaka theory. Many examples are provided.

Generalized Lie-Bäcklund theorem for Lie class $$\omega =1$$ ω = 1 overdetermined systems