We present necessary and sufficient conditions for a curve to be the center curve of an analytic and flat embedding of the Mobius strip (or an orientable cylinder) into euclidean 3-space. Using these conditions we extend an example by G. Schwarz into a continuous family of analytic and flat Mobius strips. This family is split into two connected components. We give a topological argument that explains this behaviour. A connection to the work of C. Chicone and N.J. Kalton on the Mobius strip embedding problem, suggests a close relation between a linking number and the total torsion of the orthogonal axes of a Mobius strip or an orientable cylinder.