Nonholonomic torses of the first kind

In the three-dimensional Euclidean space we study two-dimensional nonholonomic distributions of planes orthogonal to a vector field with zero total curvature of the first kind (they are called nonholonomic torses of the first kind). Using the Cartan method [1] and a canonical moving frame, we study geometric properties of two kinds: 1) one of the principal curvatures of the first kind differs from zero (the general case); 2) both principal curvatures of the first kind equal zero (a nonholonomic plane). The result in case 2) is obtained in a general form.