LTSpice Operational Integrator not Working
Here's an example of how to set things up. I've used the LT1800 (which is a rail to rail in/out opamp) to get the maximum range for display purposes. (It's also in the library.)

You can set up your voltage rails using separate voltage supplies with drawn images on the schematic. But it really isn't necessary. As you see below, it's just three spice lines. So that's another way to go. (I usually start with 99 and work the numbers backwards as I go in order to avoid LTspice's desire to start with 1 and work upwards from there as you add pictorial voltage sources to a schematic.)



As you can see above, the rate of change in the voltage at \$V_\text{OUT}\$ is \$\frac{\text{d}V_\text{OUT}}{\text{d}t}=\frac{5\:\text{V}-\left(-5\:\text{V}\right)}{500\:\text{ms}-0\:\text{ms}}=20\:\frac{\text{V}}{\text{s}}\$.

Since you know \$I=\frac{2\:\text{V}-0\:\text{V}}{10\:\text{k}\Omega}=200\:\mu\text{A}\$ then it follows that \$\frac{\text{d}V_C}{\text{d}t}=\frac{I}{C}=\frac{200\:\mu\text{A}}{10\:\mu\text{F}}=20\:\frac{\text{V}}{\text{s}}\$.

And that is a match.



The .IC card is relatively important if you want to get the largest range. What it does is sets up constraints on the solver for the initial conditions that it is forced to apply before starting the run. By setting \$V_\text{OUT}=5\:\text{V}\$, I'm forcing it to work out the fact that there will be, initially, \$5\:\text{V}\$ across capacitor \$C_1\$. 

Note that the .TRAN card uses "UIC" in order to get the run started correctly. The use of UIC means that Spice will not go through the "initial transient solution" step (the so-called "ITS") to find the DC solution at \$t=0\$.

When you use UIC, the initial value of every single energy storage (voltage and current) device is treated as zero, except for those which are explicitly provided using the .IC card. The .IC card can specify node voltages or else inductor currents.

When you don't use UIC, then everything you explicitly provide using the .IC card, gets added as a constraint. But the ITS is still permitted to otherwise find initial values based upon a steady state solution... consistent with the given constraints.

This last bit may not be what you want here. So I recommend using UIC.

If you know, a priori, all of the initial values for the energy storage devices in the circuit, you can use UIC to compute the steady state solution without the transient response leading up to it (that may occur if you instead allowed Spice to first perform the ITS step by not using UIC.)



Hopefully, you are able to interpret the above information into your situation with success.