Possible generalizations of the Minagawa–Rado Lemma on the rigidity of a surface of revolution with a fixed parallel

A general necessary and sufficient criterion is established for the rigidity of a surface of revolution $S\in C^1$ under the condition of a fixed parallel. Also two simple sufficient criteria for this property are given. It is shown by an example that this property does not hold for $S\in C^1$ in the general case.