In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

中文翻译：

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框架曲面族的包络和一阶偏微分方程的奇异解

为了研究奇异曲面的包络，我们分别介绍了框架曲面的一参数族和二参数族以及基本不变量。利用基本不变量，给出了框架曲面一参数族和二参数族的存在性和唯一性定理。然后我们定义了框架曲面的一参数族和二参数族的包络，并给出了包络的存在条件，称为包络定理。作为包络定理的一个应用，我们证明了完全可积的一阶偏微分方程的奇异解的投影是包络。