The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix $$ \left\|\frac{\partial^2u}{\partial x^i\partial x^j}\right\|\, . $$ For instance, with $n=2$, one has $$ A+Bu_{xx}+Cu_{xt}+Du_{tt}+E(u_{xx}u_{tt}-u_{xt}^2)=0\, ,\quad\quad (^*) $$ where $A,B,C,D,E$ depend on $x,t,u,u_x,u_t$.

Let's stick with the case $n=2$.

DEFINITION: A second-order bidimensional PDE is of

Monge-Ampère typeiff it can be brought in the form $(^*)$ by means of a contact transformation of the space $\{ x,t,u,u_x,u_t \}$.

Now, my question.

QUESTION: Is there an example of a second-order bidimensional PDE,

which is a polynomial of degree not greater than 2 in the second-order derivatives, and is NOT of Monge-Ampere type?

What kind of answer do I expect? It would be really great if someone pointed out some "famous" equation (in the sense that it enjoys some popularity, possibly amongst physicists), like $$ u_t=u_{xx}\, ,\quad\quad (^{**}) $$ and provided me with a very simple argument showing that $(^{**})$ cannot be brought in the form $(^{*})$. (I simply don't believe that all the equations from my question are of Monge-Ampère type - yet this feeling may be wrong.)

**COMMENTS after R. Bryant answer**

To be more precise, my question is about hypersurfaces in second-order jets spaces or, in a fibre-wise perspective, about *hypersurfaces in the Lagrangian Grassmannian*. Of course, Monge-Ampère equations form an invariant class, as they correspond to **hyperplane sections**. So, I'm interested to know what lies beyond the class of hyperplane sections: intuitively, it should be the class of **"hyperquadric" sections**, and Bryant's "cylindrical" example
$$
u_{xx}^2+u_{tt}^2=1
$$
falls indeed in such a class. **Has this class been studied?** Is there any "famous" example (e.g., describing the geometry of some special surfaces, or modeling some physical phenomenon) of a "hyperquadric" section which is not a hyperplane section?