  We explicitly construct the rank one primitive Stark (equivalently,
Kolyvagin) system extending a constant multiple of Flach's zeta elements for
semi-stable elliptic curves. As its arithmetic applications, we obtain the
equivalence between a specific behavior of the Stark system and the minimal
modularity lifting theorem, and we also discuss the cyclicity of the adjoint
Selmer groups. This Stark system construction yields a more refined
interpretation of the collection of Flach's zeta elements than the "geometric
Euler system" approach due to Flach, Wiles, Mazur, and Weston.
