I use TL082 as comparator with negative supply (-Vcc) connected to ground directly.
In the datasheet for the TL082, under
APPLICATION HINTS, it clearly states:
However, neither of the input voltages should be allowed to exceed the negative supply as this will cause large currents to flow which can result in a destroyed unit.
Your integrating opamp,
IC(2/2), is using a bipolar supply, and the output signal swings both positive and negative. However, when you connect this signal to the third opamp, which has VEE connected to ground, the signal gets clamped at ground. This is because when it tries to go below ground, the third opamp allows a "large current to flow" (i.e., its input impedance drops dramatically). This stops the oscillation, because now the first opamp,
IC(1/2), can't switch at -3V like it normally does.
The solution is to connect VEE of the third opamp to the same negative supply as the other opamps. Why didn't you do this to begin with?