A question on making a simple split power supply

A relatively painless approach, assuming you have just the requisite positive supply, say +9 Volts, would be the use of a switched-capacitor voltage converter such as the MAX1044 or ICL7660 in voltage inverting mode. The first typical operating circuit schematic in the linked datasheet shows how.

This will provide an inverted voltage supply of up to 10mA, matching the positive voltage supply, up to 10 Volts: Sufficient for XR2206 operation with a high impedance load.

For higher current requirements, other negative voltage generation modules exist from various manufacturers, typically switch mode converters.

If a supply of twice the required minimum voltage is available, another option is to simply split the supply and create a virtual ground, either using an op-amp fed from a pair of matched resistors as voltage divider, or using a dedicated rail splitter part such as the Texas Instruments TLE2426.

This article provides some ideas on virtual ground / rail splitter circuits.

No, this won't work to give you +/- supplies, because of the common negative rail.

You can, however, use the transformer you have to give you +/- adjustable supplies by using two half-wave rectifiers (2 diodes), double up on the capacitor values to keep the ripple at ~2Vp-p, and an LM317 for the positive and an LM337 for the negative supplies. With suitable (large) heat sinks capable of dissipating about 10W each it should be good for about +/-0.6A. You could also use 10,000uF caps (say 25V rated) to get a bit more voltage range at high current.

^{simulate this circuit – Schematic created using CircuitLab}