Is there a general rule on how to transform unbalanced circuits to balanced ones?
Usually you just add a balanced receiver circuit that has an unbalanced output to the front of an unbalanced circuit.
You need to understand the signal levels you want to handle on input, and the maximum signal level you want on the output.
The usual approach is to use +15V and -15V power supplies because that is a standard voltage range for op-amps, and can handle almost any signal level.
The ThatCorp receivers are very good quality.
This TI part is less expensive and with fewer external components at the cost of common mode rejection that is not as good.
TI INA1651 balanced receiver.
I have not used that specific TI part, but I built the same circuit out of separate components, a dual op-amp for input buffer, and an INA137 for the differential amp stage. The INA1651 combines the input buffers, the differential amp, and the input bias resistors all in one part.
One thing to consider with the '1651 device is that it is unity gain. With separate components you could use the '137 diff amp, which is -6dB gain, so in the case of symmetrical output swing feeding the input you can handle higher signal levels. That is a concern if you want to handle 24 dBu inputs, but if you only need to handle up to 20 dBu the '1651 can output that running on 15V rails.
You could just nudge up to 24 dBu if you use 18V rails, so it goes back to understanding what levels you need or want to handle.
For the output you can either use impedance balanced outputs as Ian mentioned, or use one of the That Corp drivers, or just buffer with a dual op-amp, one configured as a unity gain buffer and the other configured as an inverting unity gain buffer.
The dual op-amp approach really only has an advantage of increased output level swing from the symmetrical output signals. So again, back to understanding the maximum output level you need. If you only need up to +20 dBu then probably impedance balanced output is the best choice.
For best noise performance having all the circuitry run from the same power supply is the best approach. If for some reason the circuit cannot run from higher supply voltages you would have to use a voltage divider to reduce the signal level at the output of the balanced receiver, and amplify again at the output of the circuit (assuming you wanted professional line level in and out, as opposed to microphone levels for example).
There will sometimes be an advantage to padding down at the input anyway, e.g. if the circuit could generate a boost in signal level you may want to accommodate that up front, rather than making the user reduce the signal level to prevent clipping.
If some of the terminology doesn't make sense yet, the first thing is to get comfortable with dB conversions. That is a way of using logarithms to compare levels, and balanced interfaces are commonly referenced in dBu, which references to 0.775V.
For maximum input and output levels check the datasheets for the parts, they usually reference the maximum output level related to the power supply (e.g. Vs+ - 0.8V, Vs- + 1V would mean that the output can go within 0.8V of the positive supply, and within 1V of the negative supply).