## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank' and 'hours worked by Mary', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $8.62x_1^2 + 7.29x_1x_2 + 1.66x_2^2 + 9.21x_1 + 9.73x_2$.

## 3: List the constraints in symbolic notation
The constraints are:
- $3x_1 \geq 0$ (Hank's paperwork competence rating, but this is inherently satisfied as $x_1 \geq 0$)
- $6x_1 \geq 0$ (Hank's work quality rating, but this is inherently satisfied as $x_1 \geq 0$)
- $13x_1 \geq 0$ (Hank's computer competence rating, but this is inherently satisfied as $x_1 \geq 0$)
- $13x_2 \geq 0$ (Mary's paperwork competence rating, but this is inherently satisfied as $x_2 \geq 0$)
- $3x_2 \geq 0$ (Mary's work quality rating, but this is inherently satisfied as $x_2 \geq 0$)
- $13x_2 \geq 0$ (Mary's computer competence rating, but this is inherently satisfied as $x_2 \geq 0$)
- $3x_1 + 13x_2 \geq 13$ (total combined paperwork competence rating)
- $6x_1 + 3x_2 \geq 22$ (total combined work quality rating)
- $13x_1 + 13x_2 \geq 16$ (total combined computer competence rating)
- $-9x_1 + 4x_2 \geq 0$
- $3x_1 + 13x_2 \leq 19$ (total combined paperwork competence rating upper bound)
- $6x_1 + 3x_2 \leq 63$ (total combined work quality rating upper bound)
- $(13x_1)^2 + (13x_2)^2 \leq 34^2$ is not correct based on the problem description. The correct constraint is $13x_1 + 13x_2 \leq \sqrt{34}$ is also not correct. The problem states that "The total combined computer competence rating from hours worked by Hank squared plus hours worked by Mary squared must be 34 at maximum." So, $(13x_1)^2 + (13x_2)^2 \leq 34$ 
- $x_2$ must be an integer.

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

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(lb=0, name="hours_worked_by_Hank")  # hours worked by Hank
x2 = m.addVar(lb=0, type=gurobi.GRB.INTEGER, name="hours_worked_by_Mary")  # hours worked by Mary

# Objective function
m.setObjective(8.62*x1**2 + 7.29*x1*x2 + 1.66*x2**2 + 9.21*x1 + 9.73*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(3*x1 + 13*x2 >= 13, name="paperwork_competence_rating")
m.addConstr(6*x1 + 3*x2 >= 22, name="work_quality_rating")
m.addConstr(13*x1 + 13*x2 >= 16, name="computer_competence_rating")
m.addConstr(-9*x1 + 4*x2 >= 0, name="labor_hours_constraint")
m.addConstr(3*x1 + 13*x2 <= 19, name="paperwork_competence_rating_upper_bound")
m.addConstr(6*x1 + 3*x2 <= 63, name="work_quality_rating_upper_bound")
m.addConstr((13*x1)**2 + (13*x2)**2 <= 34, name="computer_competence_rating_squared_upper_bound")

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Hank: {x1.varValue}")
    print(f"Hours worked by Mary: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Mary')],
    'objective_function': '8.62*x1^2 + 7.29*x1*x2 + 1.66*x2^2 + 9.21*x1 + 9.73*x2',
    'constraints': [
        '3*x1 + 13*x2 >= 13',
        '6*x1 + 3*x2 >= 22',
        '13*x1 + 13*x2 >= 16',
        '-9*x1 + 4*x2 >= 0',
        '3*x1 + 13*x2 <= 19',
        '6*x1 + 3*x2 <= 63',
        '(13*x1)^2 + (13*x2)^2 <= 34',
        'x2 is an integer'
    ]
}
```