Draw the charge configuration according to the electric field I'm trying to draw the charge configuration outside the regions that lead to the given electric field patterns. Can someone show me how?

 A: Using cylindrical coordinates with the origin at the center and the $\phi = 0$ direction 'down' (the OP says the image should be rotated CCW 90 degrees), the electric field appears be have only a radial component with a sign change for $\phi = \frac{-\pi}{2}$ and $\phi = \frac{\pi}{2}$
$$\vec E = E(\rho,\phi)\hat\rho $$
$$E(\rho,\phi) = E_{\rho}(\rho),\quad -\frac{\pi}{2} \lt\phi \lt \frac{\pi}{2}$$
$$E(\rho,\phi) = -E_{\rho}(\rho),\quad \frac{\pi}{2} \lt \phi \lt \frac{3\pi}{2}$$
The $z$ component of the curl of this field is thus
$$(\nabla \times \vec E)\cdot \hat z = -\frac{1}{\rho}\frac{\partial E(\rho,\phi)}{\partial \phi}$$
This is zero except at $\phi = \pm \frac{\pi}{2}$ where it is 'infinite' at the discontinuity.
You can see this in the picture without resorting to math.  Along the horizontal line, the field lines above the line are opposite those below the line.  A closed line integral of the field that goes above and below the line will be non-zero.
Thus, since the field is not irrotational, this cannot be the field of a static charge configuration.
A: I have came up with this:
Charges are the sources of the electric field. So, whatever the point that field lines are "created" or "destroyed", must be a charge. Then, if there are a charge, then must be on the center.
Calculating the electric flux:
$$ 
\phi = \iint_S\ \mathbf E\cdot d\mathbf s = \frac{Q}{\epsilon_0}
$$
Let's pick a sphere as gaussian surface. From the draw, seems reasonable to assume $\phi = 0$, since half of field lines are entering, and half are leaving. If there's no flux, there's no charge.
Then $Q = 0$. There are "two" possibilities: Equal charges $+q$ and $-q$ forming up infinitesimal dipole, or no charges. Since this field configuration is not dipole, then must have no charges.
Of course could be more complex configuration of charges, like "quadrupoles" and so on. This are not possible too, because the field lines are being created and destroyed in the same point, without existance of a "symmetry line".
If there's no charge, there's no field lines. Then, this field configuration seems to be impossible.
