“…Let be the part of the dual solution associated with the negative eigenvalue α j , so that Following the construction of dual singularities introduced in , it appears here that the dual singularities , which are associated with an integer eigenvalue , contain logarithmic terms in ρ . In this paper, we make the following ansatz for these dual singularities: and hereafter, we provide explicit recursive ODEs for the determination of the corresponding polar parts ψ j , ℓ ( ϕ )and .…”