Let $f(x,y) \in \mathbb{R}^2$ that belongs to $C^{\infty}$-class, denote $H = \begin{bmatrix} \frac{\partial^2f(x,y)}{\partial x^2} & \frac{\partial^2f(x,y)}{\partial x\partial y} \\ \frac{\partial^2f(x,y)}{\partial x\partial y} & \frac{\partial^2f(x,y)}{\partial y^2} \end{bmatrix}$.
The picture below illustrates the gradients of $f(x, y)$ with the step of $0.5$. The vector comes from the point where gradient is calculated.
It's claimed that there exists a straight line (more than one) along which $H$ is singular. Find every line and its slope $a$ (as in $y = ax + b$).
Is it true that there are points where the gradient would not be equal to zero, but starting from which it's impossible to reach the minimum using a gradient descent?
I'd appreciate the hints. 