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We define an analogue of the Arnol'd surface for odd degree flexible curves,and we use it to double branch cover totally flexible embeddings, where totallyflexible is a simple and natural condition to be added to the classical notionof a flexible curve. This allows to obtain a Viro--Zvonilov type inequality: anupper bound on the number of non-empty ovals of a curve of odd degree. We alsodigress around a possible definition of non-orientable (totally) flexiblecurves, for which our method still works and our inequality almost holds. Weinvestigate our method for flexible curves in a quartic to derive a similarbound in two cases.

A Viro-Zvonilov type inequality for totally flexible curves of odd degree