## Step 1: Define the symbolic representation of the variables
The variables are 'light infantry companies' and 'mechanized infantry companies', which can be represented symbolically as $x_0$ and $x_1$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $5.17x_0 + 6.54x_1$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $4x_0 + 11x_1 \geq 14$ (minimum logistics footprint)
- $5x_0 + 11x_1 \geq 20$ (minimum logistical capacity)
- $8x_0 + 8x_1 \geq 17$ (minimum defensive capability rating)
- $2x_0 - x_1 \geq 0$ (relationship between light infantry and mechanized infantry companies)
- $4x_0 + 11x_1 \leq 31$ (maximum logistics footprint)
- $5x_0 + 11x_1 \leq 64$ (maximum logistical capacity)
- $8x_0 + 8x_1 \leq 31$ (maximum defensive capability rating)
- $x_0 \geq 0$ and $x_1 \geq 0$ (non-negativity constraints)
- $x_0$ and $x_1$ are integers (integer constraints)

## 4: Convert the problem into a Gurobi model
We will use Gurobi to solve this mixed-integer linear programming problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("infantry_companies")

# Define the variables
x0 = m.addVar(name="light_infantry_companies", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="mechanized_infantry_companies", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(5.17 * x0 + 6.54 * x1, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(4 * x0 + 11 * x1 >= 14, name="min_logistics_footprint")
m.addConstr(5 * x0 + 11 * x1 >= 20, name="min_logistical_capacity")
m.addConstr(8 * x0 + 8 * x1 >= 17, name="min_defensive_capability")
m.addConstr(2 * x0 - x1 >= 0, name="infantry_relationship")
m.addConstr(4 * x0 + 11 * x1 <= 31, name="max_logistics_footprint")
m.addConstr(5 * x0 + 11 * x1 <= 64, name="max_logistical_capacity")
m.addConstr(8 * x0 + 8 * x1 <= 31, name="max_defensive_capability")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Light Infantry Companies: {x0.varValue}")
    print(f"Mechanized Infantry Companies: {x1.varValue}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'light infantry companies'), ('x1', 'mechanized infantry companies')],
    'objective_function': '5.17*x0 + 6.54*x1',
    'constraints': [
        '4*x0 + 11*x1 >= 14',
        '5*x0 + 11*x1 >= 20',
        '8*x0 + 8*x1 >= 17',
        '2*x0 - x1 >= 0',
        '4*x0 + 11*x1 <= 31',
        '5*x0 + 11*x1 <= 64',
        '8*x0 + 8*x1 <= 31',
        'x0 >= 0',
        'x1 >= 0'
    ]
}
```