On the road to N=2 supersymmetric Born-Infeld action

We analyze the exact perturbative solution of N=2 Born-Infeld theory which is believed to be defined by Ketov's equation. This equation can be considered as a truncation of an infinite system of coupled differential equations defining Born-Infeld action with one manifest N=2 and one hidden N=2 supersymmetries. We explicitly demonstrate that infinitely many new structures appear in the higher orders of the perturbative solution to Ketov's equation. Thus, the full solution cannot be represented as a function depending on {\it a finite number} of its arguments. We propose a mechanism for generating the new structures in the solution and show how it works up to 18-th order. Finally, we discuss two new superfield actions containing an infinite number of terms and sharing some common features with N=2 supersymmetric Born-Infeld action.