Can integer linear programs have non-integer OPT solutions?
Something that might be causing you confusion: "integer linear programming", by definition, restricts the variables to integer values.
Linear programming instances in which the input numbers are integers can easily have only non-integer solutions. Consider:

$\begin{cases}
\begin{alignat}{1}
2 &amp;\cdot x \le 1
\\[1.5ex]
-2 &amp;\cdot x \le -1
\end{alignat}
\end{cases}$