Do not attempt to repair the device yourself if you want to retain your warranty. If you contact Sony they will send you a box and you can ship it to them for repair. If you are outside of your warranty then you can look into replacing your power supply. Usually the power supply will pop and short out if it is full of dust, and a ESD event causes damage. Dust is made up of mostly human skin and human skin has oil which can carry a charge across the circuit board causing a short.
Many laptop manufacturers run the LCD cable through the hinges. The fit is usually snug, and over time the cable will pull on the connector which attaches at the mother board. Take the bottom off and look for a braided black cable that connects to the Mother Board. Make sure that the cable is not frayed, and can snap tightly into the motherboard. If it is snug then your cable has pulled lose from the screen and ou will need to take it in for service. The reason it flickers when you rock it back and forth is because the cable becomes tighter when you lean the screen in the backwards position. Opening a screen by using the center of the cover rather then a corner will greatly reduce the wear and tear on the hinges.