Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996). An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
For more details you can see the following interesting papers:
1) Zariski Geometries, Zilber and Hrushovski.
2) Analytic Zariski structures and the Hrushovski construction, Peatfield and Zilber.
Question : Under what conditions an analytic Zariski structure is stable ?