Polynomial Integrals of the Geodesics Equations in Two-Dimensional Case

Let $M^2$ be two-dimensional surface with the Riemannian metric
$$ \tag{1} ds^2=g_{11}(x,y)dx^2+2g_{12}(x,y)dxdy+g_{22}(x,y)dy^2. $$
Geodesics equations of a given metric can be treated as a system of Euler-Lagrange equations
$$ \tag{2} \frac{d}{dt}L_{\dot{x}}-L_x=0,\qquad \frac{d}{dt}L_{\dot{y}}-L_y=0 $$
with the Lagrangian
$$ \tag{3} L(x,y,\dot{x},\dot{y})=\frac 12g_{11}(x,y)\dot{x}^2 +g_{12}(x,y)\dot{x}\dot{y}+\frac 12g_{22}(x,y)\dot{y}^2. $$
Geodesic flow of the metric (1) is Liouville integrable, if it possesses a smooth first integral $F$ functionally independent of the Lagrangian (3). In the present work we consider the problem of the existence of the polynomial integral of the system (2), (3) of the first degree
$$ \tag{4} F_1=b_0(x,y)\dot{x}+b_1(x,y)\dot{y}, $$
the second degree
$$ \tag{5} F_2=b_0(x,y)\dot{x}^2+2b_1(x,y)\dot{x}\dot{y}+b_2(x,y)\dot{y}^2 $$
and the third degree
$$ \tag{6} F_3=b_0(x,y)\dot{x}^3+3b_1(x,y)\dot{x}^2\dot{y}+3b_2(x,y)\dot{x}\dot{y}^2+b_3(x,y)\dot{y}^3. $$
For an integral (4), (5) or (6) the existence conditions are obtained as the compatibility conditions of an overdetermined system of linear homogeneous first-order equations in the functions $b_i(x,y)$. Here these conditions are expressed in terms of the invariants
$$ \tag{7} I_1(x,y)=\frac{J_1}{j_0J_0^3},\qquad I_2(x,y)=\frac{J_2}{j_0J_0^2} $$
of the equivalence transformations of the family of equations (2), (3) defined by
$$ \tilde t=k(t+t_0),\qquad \tilde x=\varphi(x,y),\qquad \tilde y=\psi(x,y),\qquad k,t_0={\rm const}. $$
In (7) the value $J_0$ up to a constant multiplier coincides with the main (scalar) curvature $K$ of the surface $M^2$,
$$ j_0=g_{11}g_{22}-g_{12}^2,\qquad J_1=g_{22}J_{0x}^2-2g_{12}J_{0x}J_{0y}+g_{11}J_{0y}^2. $$

All results on the integrals (4)–(6) are obtained in assumption of the non-degeneracy of the surface $M^2$. It means, when the conditions
$$ \tag{8} j_0\neq 0,\qquad J_0\neq 0,\qquad J_1\neq 0 $$
hold. The geometrical sense of the first two conditions (8) is obvious (non-degeneracy of the matrix $g_{ij}(x,y)$ and nonzero curvature of the surface). The sense of the third condition (8) is not so evident. The question is what properties has the degenerate surface $M^2$ with the curvature $K$, which satisfies the relation
$$ \tag{9} g_{22}(x,y)\left(\frac{\partial K}{\partial x}\right)^2 -2g_{12}(x,y)\frac{\partial K}{\partial x}\frac{\partial K}{\partial y} +g_{11}(x,y)\left(\frac{\partial K}{\partial y}\right)^2=0. $$