Op Amp integrator - limiting +Vsat to zero (ground)
@Tim's suggestion is a good one. An analog switch across the integrating capacitor with the inputs grounded. Note that it's usually better to use the same resistor for integrate and de-integrate, though if  your switch has a lot of charge injection it might cause issues. Ideally the value of the resistor, the capacitor value and the absolute clock frequency will cancel out. 

You might find the front end of a typical ca. 1990 DVM chip of interest: 



In the auto-zero (AZ) phase, the comparator output is fed back to the integrator input. The differential inputs are shorted and grounded. This causes oscillation at the comparator output but it results in charging Caz to compensate for the offset voltage of the integrator and input buffer, meaning a lot better precision for the relatively poor performance of those op-amps. Typically it will auto-zero within tens of uV. 



By including a few extra switches they allow a fully differential floating reference voltage via a "flying capacitor" Cref.