A generalized $MV$-algebra $𝒜$ is called representable if it is a subdirect product of linearly ordered generalized $MV$-algebras. Let $S$ be the system of all congruence relations $\rho$ on $𝒜$ such that the quotient algebra $𝒜/\rho$ is representable. In the present paper we prove that the system $S$ has a least element.