LM319 can accept negative supply, eg power rails up to ±18 V or +11 V to -25 V (with ground at 0V)
Within those (and other) parameters, it doesn't matter what absolute values you assign the three voltages (at least until you connect to another circuit!)
As far as I can see from the datasheet, the output ranges between approx Vs+ and ground.
Vs- can be ground or negative of ground.
There is no law that says ground has to be your 0 V reference, so you could have Vs+ as say +5 V, gnd as -5 V and Vs- as -6 V to achieve an output roughly symmetrical about 0 V. (Only approx as the output does not reach exactly the power rail nor ground.
But you cannot reach peak to peak output of 20 V (±10 V) because that would exceed the maximum rating of 16 V between ground and Vcc.
You could follow the oscillator with a buffer stage, whose job was simply to amplify the signal and limit the output to ±10 V. That would require a different op amp. But then you could just use that alone as the osc., if you only wanted a rough output !
So for what you ask, I think I'd use a different op amp.
