M-curves of degree 9 with four or three nests

We consider now M -curve of degree 9 with real scheme 〈J q 1〈α1〉 q 1〈α2〉 q 1〈α3〉 q 1〈α4〉 q β〉 or 〈J q 1〈α1〉 q 1〈α2〉 q 1〈α3〉 q β〉. In this chapter, we call a nest a piece of curve 1〈αi〉, and the curves under consideration have either four or three nests. Let Oi, i = 1, . . . 3 or 4 be the non-empty ovals. For convenience, we shall sometimes name a nest after its exterior oval Oi.