On societies of characteristic surfaces in the space of two complex variables. II: Degenerative essential singularity. (Q1175052)

[Part I of this paper in J. Math. Soc. Japan 23, 53--81 (1971; Zbl 0203.39002); for a survey see Sci. Rep. Kyoto Pref. Univ. 38, A1--A3 (1987)] The author furthers the investigations of the society \(S\) in a subdomain \(D\) of the product space of two Riemann spheres. Let \(E\) be a degenerative set in \(D\) and assume that \(S\) is globally analytic and that at least one point of \(E\) is an essential singularity of \(S\). In the present paper, the author investigates its structure and proves that the set \(E^*\subset E\) of essential singularities of \(S\) is a degenerative set in \(D\backslash E\) and that the analytic projection of \(S\) is analytically equivalent to either the Riemann sphere, the complex plane \(\mathbb{C}\), \(\mathbb{C}\backslash\{0\}\) or the quotient space of \(\mathbb{C}\backslash\{0\}\) by a discrete multiplicative group \(\{\exp(2\pi\lambda j)\): \(j\) an integer), \(\lambda\) being a complex number with positive real part. He also discusses ordinary differential equations of first order associated with this problem.