Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution

Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heuns equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example where one encounters a form of the Heun equation.