A center fed dipole, with equal arm lengths, is a balanced device.
If it is fed using a gamma match - does it change (the system) to be unbalanced?
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A half wave wire has a low impedance point at its midpoint. If you put two half wave wires at 90 degrees with respect to each other so the midpoints come close, but not in electrical contact with each other you can measure the impedance between the midpoints. You will find that it is very low.
Each half wave can be seen as two quarter waves. When you feed two quarter wave rods in phase, the radiation from both will be very similar, but 180 degrees out of phase so the radiation will cancel. A quarter wave will transform the very high impedance at one end (the tip) to a low impedance at the other end. Since there is almost no radiation the impedance at the tip becomes very high and the impedance at the midpoint very low.
The secret of the gamma match is that you use the virtual ground point at the midpoint to connect the screen of your feed cable. It is the same phenomenon we use in ground plane antennas.
Actually half wave rods can be used to place ground points on support structures and cables along them for example when using X-yagis. Such a virtual ground point should typically be parallel to the boom tube, but placement is critical, one wants the virtual ground point to create a high impedance point where the support structure connects to the boom tube typically 3/4 wl from the boom tube (1/4 wl is too close.)
Once the screen is connected to a virtual ground point you can always match the impedance by a suitable LC link. The current on your coax screen will be small. In a yagi where the element length is significantly different from 0.5 wl it is a good idea to connect the midpoint as well as the screen to the boom tube to further bring down the impedance.
The system says I need a 50 reputation to add a comment, which I don't have yet so I'll add it as a post. The short answer to the original question is, the original question needs to have some terms defined because there are some contradictions: