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 pseudo spherical surface determined by a solution a of such an equation, do the coefficients of the second fundamental form of the local isometric immersion in R-3 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 irrirrwrsion exists, the coefficients are universal functions of x and t, independent of u.