What are the ways to make a dual power supply from a single voltage source?

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.

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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.

For a quick and simple dual power supply, use two resistors in series connected in parallel with two capacitors. Connect the two ends to the battery or power source and BAM! You have a dual power supply. Typical values for bipolar converters like this are 100k-1M for the resistors and 47uf to 4700uf depending on the current draw of your circuit.

^{simulate this circuit – Schematic created using CircuitLab}

Note:!!! Make sure the negative rail of your power supply is not connected to ground! Use the continuity setting on your DMM to check !!!

The XR2206 works just fine off a single supply without any kind of rail splitting, you just need to read the datasheet. Figure 2 shows a single supply example.

If you go this route, be mindful of the minimum single supply voltage.

Also, a word of caution: the XR2206 is several years obsolete. Do not use it for anything that you may want to put into production.