What happen if we put permanent potential across conductor? why if we put conductors under permanent potential like a battery ,it will not be equi-potential ??
and why if we just put it under electric field it is equi-potential although there is a surface charges positive and negative ?
 A: You will force a voltage across the conductor in series with the internal resistor of the battery and since it represents a very very low resistance, a huge current will flow (see Ohm's law), eventually heating up the wire by Joule's law, possibly burning/melting it. Though the battery will be the first one to suffer, and it can be dangerous for some types (that's why usually those ones have internal protections, but still). 
Indicentally, that's what happens when you put in parallel two batteries, even if they have the same nominal voltage. They never really do, so you end up with the same problem of a non-negligible voltage across a wire.
A: The two sides will eventually become equipotential. At that point, we say the battery has run dry. The point is that its not instantaneous; it takes time for the current to move charge from one end of the battery to the other until the battery loses its potential difference. 
The potential from the battery is in no way permanent. If the potential was permanent, an assumption demanded the title of your question, then by definition it would stay permanent and never become an equipotential. Like the two prongs in the socket of my wall - they are always at a potential no matter how long my cell phone charger is plugged in. Of course, this isn't for free - it's because the electrical grid and power plants do a good job of running continuously.
Your second paragraph doesn't make any sense. If there are positive and negative surfaces charges, in equal amounts and on infinite planes, there is not an equipotential. The potential is strongest near the two surfaces and switches sign somewhere in the middle. The electric field, the gradient of this potential, is constant.
