## Step 1: Define the symbolic representation of the variables
The variables are 'mechanized infantry companies' and 'medical platoons', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the symbolic representation of the objective function
The objective function to maximize is $9x_1^2 + 6x_1x_2 + 2x_2^2 + x_2$.

## Step 3: Define the symbolic representation of the constraints
The constraints are:
- $12x_1 + 1x_2 \leq 133$ (offensive capability rating)
- $18x_1 + 15x_2 \leq 162$ (logistical capacity)
- $14x_1 + 9x_2 \leq 68$ (fun factor)
- $17x_1 + 11x_2 \leq 95$ (defensive capability rating)
- $12^2x_1^2 + 1^2x_2^2 \geq 22$ (total combined offensive capability rating)
- $18x_1 + 15x_2 \geq 40$ (total combined logistical capacity)
- $14x_1 + 9x_2 \geq 26$ (total combined fun factor)
- $17^2x_1^2 + 11^2x_2^2 \geq 29$ (total combined defensive capability rating)
- $-7x_1 + 6x_2 \geq 0$ (linear constraint)
- $12x_1 + 1x_2 \leq 102$ (total combined offensive capability rating)
- $18x_1 + 15x_2 \leq 107$ (total combined logistical capacity)
- $14x_1 + 9x_2 \leq 42$ (total combined fun factor)
- $17x_1 + 11x_2 \leq 63$ (total combined defensive capability rating)
- $x_1 \geq 0, x_1 \in \mathbb{Z}$ (mechanized infantry companies are non-negative integers)
- $x_2 \geq 0, x_2 \in \mathbb{Z}$ (medical platoons are non-negative integers)

## Step 4: Convert the problem into a Gurobi code
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="mechanized_infantry_companies", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="medical_platoons", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(9*x1**2 + 6*x1*x2 + 2*x2**2 + x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(12*x1 + x2 <= 133, name="offensive_capability_rating")
m.addConstr(18*x1 + 15*x2 <= 162, name="logistical_capacity")
m.addConstr(14*x1 + 9*x2 <= 68, name="fun_factor")
m.addConstr(17*x1 + 11*x2 <= 95, name="defensive_capability_rating")
m.addConstr(12**2*x1**2 + x2**2 >= 22, name="total_offensive_capability_rating")
m.addConstr(18*x1 + 15*x2 >= 40, name="total_logistical_capacity")
m.addConstr(14*x1 + 9*x2 >= 26, name="total_fun_factor")
m.addConstr(17**2*x1**2 + 11**2*x2**2 >= 29, name="total_defensive_capability_rating")
m.addConstr(-7*x1 + 6*x2 >= 0, name="linear_constraint")
m.addConstr(12*x1 + x2 <= 102, name="total_offensive_capability_rating_2")
m.addConstr(18*x1 + 15*x2 <= 107, name="total_logistical_capacity_2")
m.addConstr(14*x1 + 9*x2 <= 42, name="total_fun_factor_2")
m.addConstr(17*x1 + 11*x2 <= 63, name="total_defensive_capability_rating_2")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Mechanized Infantry Companies: ", x1.varValue)
    print("Medical Platoons: ", x2.varValue)
else:
    print("The model is infeasible")
```

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'mechanized infantry companies'), ('x2', 'medical platoons')],
    'objective_function': '9*x1^2 + 6*x1*x2 + 2*x2^2 + x2',
    'constraints': [
        '12*x1 + x2 <= 133',
        '18*x1 + 15*x2 <= 162',
        '14*x1 + 9*x2 <= 68',
        '17*x1 + 11*x2 <= 95',
        '12^2*x1^2 + x2^2 >= 22',
        '18*x1 + 15*x2 >= 40',
        '14*x1 + 9*x2 >= 26',
        '17^2*x1^2 + 11^2*x2^2 >= 29',
        '-7*x1 + 6*x2 >= 0',
        '12*x1 + x2 <= 102',
        '18*x1 + 15*x2 <= 107',
        '14*x1 + 9*x2 <= 42',
        '17*x1 + 11*x2 <= 63',
        'x1 >= 0, x1 in Z',
        'x2 >= 0, x2 in Z'
    ]
}
```