In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.
Andrew D. Hwang. Xinyi Wang. "Clairaut surfaces in Euclidean three-space." Tohoku Math. J. (2) 74 (2) 215 - 227, 2022. https://doi.org/10.2748/tmj.20201209