## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'air defense batteries' and 'artillery batteries', which we can denote as $x_1$ and $x_2$ respectively. The objective function to maximize is $7x_1^2 + 6x_1x_2 + 6x_2^2 + 3x_2$. The constraints are:
- $12x_1 \leq 154$ (defensive capability rating of air defense batteries)
- $12x_1 \leq 93$ (deployment weight of air defense batteries)
- $18x_2 \leq 154$ (defensive capability rating of artillery batteries)
- $1x_2 \leq 93$ (deployment weight of artillery batteries)
- $x_1^2 + x_2^2 \geq 55$ (total combined defensive capability rating)
- $12x_1 + 1x_2 \geq 21$ (deployment weight)
- $-5x_1^2 + 9x_2^2 \geq 0$
- $x_1^2 + x_2^2 \leq 90$
- $12x_1 + 18x_2 \leq 90$ (total combined defensive capability rating)
- $12x_1 + 1x_2 \leq 76$ (total deployment weight)
- $x_1, x_2$ are integers.

## Step 2: Convert the problem into a symbolic representation
The symbolic variables are:
- $x_1$ for 'air defense batteries'
- $x_2$ for 'artillery batteries'

The objective function is: $7x_1^2 + 6x_1x_2 + 6x_2^2 + 3x_2$

The constraints in symbolic form are:
- $12x_1 \leq 154$
- $12x_1 \leq 93$
- $18x_2 \leq 154$
- $x_2 \leq 93$
- $x_1^2 + x_2^2 \geq 55$
- $12x_1 + x_2 \geq 21$
- $-5x_1^2 + 9x_2^2 \geq 0$
- $x_1^2 + x_2^2 \leq 90$
- $12x_1 + 18x_2 \leq 90$
- $12x_1 + x_2 \leq 76$

## 3: Formulate the problem in Gurobi
To solve this problem using Gurobi, we need to define the variables, the objective function, and the constraints.

```python
import gurobi as gp

# Define the model
m = gp.Model()

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

# Define the objective function
m.setObjective(7*x1**2 + 6*x1*x2 + 6*x2**2 + 3*x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(12*x1 <= 154, name="def_cap_rating_air_defense")
m.addConstr(12*x1 <= 93, name="deployment_weight_air_defense")
m.addConstr(18*x2 <= 154, name="def_cap_rating_artillery")
m.addConstr(x2 <= 93, name="deployment_weight_artillery")
m.addConstr(x1**2 + x2**2 >= 55, name="total_def_cap_rating")
m.addConstr(12*x1 + x2 >= 21, name="deployment_weight_total")
m.addConstr(-5*x1**2 + 9*x2**2 >= 0, name="def_cap_rating_relation")
m.addConstr(x1**2 + x2**2 <= 90, name="total_def_cap_rating_limit")
m.addConstr(12*x1 + 18*x2 <= 90, name="total_def_cap_rating_joint_limit")
m.addConstr(12*x1 + x2 <= 76, name="total_deployment_weight_limit")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Air defense batteries: ", x1.varValue)
    print("Artillery batteries: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 4: Provide the symbolic representation in JSON format
```json
{
    "sym_variables": [
        ["x1", "air defense batteries"],
        ["x2", "artillery batteries"]
    ],
    "objective_function": "7*x1^2 + 6*x1*x2 + 6*x2^2 + 3*x2",
    "constraints": [
        "12*x1 <= 154",
        "12*x1 <= 93",
        "18*x2 <= 154",
        "x2 <= 93",
        "x1^2 + x2^2 >= 55",
        "12*x1 + x2 >= 21",
        "-5*x1^2 + 9*x2^2 >= 0",
        "x1^2 + x2^2 <= 90",
        "12*x1 + 18*x2 <= 90",
        "12*x1 + x2 <= 76"
    ]
}
```