PCP with commutative alphabet
Let  $\gamma_{ij},\delta_{ij}$ be the number of characters $\sigma_j$ in $\gamma_i,\delta_i$ (respectively). Then the PCP instance has a solution if there exist non-negative integers $a_1,\ldots,a_N,b_1,\ldots,b_N$ such that for all $j \in \{1,\ldots,K\}$,
$$
\sum_{i=1}^N a_i \gamma_{ij} = \sum_{i=1}^N b_i \delta_{ij}.
$$

While integer programming is hard in the general case, in this case we can solve it using linear programming in polynomial time. I claim that the program above has a solution in non-negative integers iff it has a solution in non-negative reals. One direction is obvious. In the other direction, suppose that the linear program corresponding to the integer program above (replacing non-negative integers with non-negative reals) is feasible. Then there is a solution for the program above in the non-negative rationals (every vertex of the polytope would do). Multiplying this solution by a common denominator, we obtain a solution of the original integer program.