Second-order equations and local isometric immersions of pseudo-spherical surfaces

We consider the class of differential equations that describe pseudo-spherical surfaces of the form u_t = F(u, u_x,u_xx) and u_xt = F(u, u_x). We answer the following question: Given a pseudospherical surface determined by a solution u of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in ℝ³ depend on a jet of finite order of u? We show that, except for the sine-Gordon equation, where the coefficients depend on a jet of order zero, for all other differential equations, whenever such an immersion exists, the coefficients are universal functions of x and t, independent of u.