  Whatever the fluid lipid vesicle is modeled as the spontaneous-curvature,
bilayer-coupling, or the area-difference elasticity, and no matter whether a
pulling axial force applied at the vesicle poles or not, a universal shape
equation presents when the shape has both axisymmetry and up-down symmetry.
This equation is a second order nonlinear ordinary differential equation about
the sine $sin\psi (r)$ of the angle $\psi (r)$ between the tangent of the
contour and the radial axis $r$. However, analytically there is not a generally
applicable method to solve it, while numerically the angle $\psi (0)$ can not
be obtained unless by tricky extrapolation for $r=0$ is a singular point of the
equation. We report an infinite series representation of the equation, in which
the known solutions are some special cases, and a new family of shapes related
to the membrane microtubule formation, in which $sin\psi (0)$ takes values from
0 to $\pi /2$, is given.
