Two planar ends between two catenoid ends (finite Riemann 2)

Weierstrass Data

\[ M=\mathbb{C}\setminus\{0,\rho,-1/\rho\}, \] where $\rho\gt 0$ is a constant. \[ g=\frac{(z-\rho)(\rho z+\alpha^{-1})(\rho z +\alpha)}{(\rho z+1)(z-\rho\alpha)(z-\rho\alpha^{-1})},\qquad \eta = \left(\frac{(z-\rho\alpha)(z-\rho\alpha^{-1})}{z(z-\rho)}\right)^2 dz, \] where $\alpha =e^{\pi i/3}$.