## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Paul' and 'hours worked by Ringo', which we can denote as $x_1$ and $x_2$ respectively. The objective function to minimize is $1.67x_1x_2 + 6.79x_2$. The constraints are as follows:
- $19x_1 + 2x_2 \geq 44$ (combined paperwork competence rating)
- $18x_1 + 7x_2 \geq 31$ (combined dollar cost per hour)
- $14x_1 + 14x_2 \geq 56$ (combined organization score)
- $13x_1 + 18x_2 \geq 46$ (combined computer competence rating)
- $6x_1 - 10x_2 \geq 0$
- $19x_1^2 + 2x_2^2 \geq 44$ is not linear, but $19x_1 + 2x_2 \geq 44$ is given, and we also have $19x_1^2 + 2x_2^2 \leq 148$ and $x_1^2 + x_2^2$ is not directly constrained but $x_1$ and $x_2$ have individual bounds from other constraints.
- $18x_1 + 7x_2 \leq 82$
- $14x_1^2 + 14x_2^2 \leq 158$
- $13x_1 + 18x_2 \leq 93$
- $x_2$ must be an integer.

## 2: Correct and Linearize the Constraints for Gurobi
Correcting and linearizing where necessary:
- The paperwork constraint is $19x_1 + 2x_2 \geq 44$ and $19x_1^2 + 2x_2^2 \geq 44$ is not directly used but we have bounds.
- Other constraints are mostly linear.

## 3: Formulate the Problem in Gurobi
We will use Gurobi to solve this problem. The problem can be formulated as:
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="hours_worked_by_Paul", lb=0)  # Non-integer allowed
x2 = m.addVar(name="hours_worked_by_Ringo", lb=0, integrality=gp.GRB.Integer)  # Integer required

# Define the objective function
m.setObjective(1.67 * x1 * x2 + 6.79 * x2, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(19 * x1 + 2 * x2 >= 44, name="paperwork_1")
m.addConstr(18 * x1 + 7 * x2 >= 31, name="dollar_cost")
m.addConstr(14 * x1 + 14 * x2 >= 56, name="organization_score")
m.addConstr(13 * x1 + 18 * x2 >= 46, name="computer_competence")
m.addConstr(6 * x1 - 10 * x2 >= 0, name="six_x1_minus_ten_x2")
m.addConstr(18 * x1 + 7 * x2 <= 82, name="dollar_cost_upper")
m.addConstr(14 * x1**2 + 14 * x2**2 <= 158, name="organization_score_upper")  # Note: This is not linear, consider using gp.quadratic_constraint
m.addConstr(13 * x1 + 18 * x2 <= 93, name="computer_competence_upper")

# The paperwork upper bound and others need to be considered
m.addConstr(19 * x1**2 + 2 * x2**2 <= 148, name="paperwork_upper")  # Non-linear

# Solve the problem
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Paul: {x1.varValue}")
    print(f"Hours worked by Ringo: {x2.varValue}")
else:
    print("No optimal solution found.")
```

## 4: Symbolic Representation
The symbolic representation is:
```json
{
    'sym_variables': [('x1', 'hours worked by Paul'), ('x2', 'hours worked by Ringo')],
    'objective_function': '1.67*x1*x2 + 6.79*x2',
    'constraints': [
        '19*x1 + 2*x2 >= 44',
        '18*x1 + 7*x2 >= 31',
        '14*x1 + 14*x2 >= 56',
        '13*x1 + 18*x2 >= 46',
        '6*x1 - 10*x2 >= 0',
        '18*x1 + 7*x2 <= 82',
        '14*x1^2 + 14*x2^2 <= 158',
        '13*x1 + 18*x2 <= 93',
        '19*x1^2 + 2*x2^2 <= 148'
    ]
}
```

The final answer is: 
```python
import gurobi as gp

m = gp.Model()

x1 = m.addVar(name="hours_worked_by_Paul", lb=0)
x2 = m.addVar(name="hours_worked_by_Ringo", lb=0, integrality=gp.GRB.Integer)

m.setObjective(1.67 * x1 * x2 + 6.79 * x2, gp.GRB.MINIMIZE)

m.addConstr(19 * x1 + 2 * x2 >= 44, name="paperwork_1")
m.addConstr(18 * x1 + 7 * x2 >= 31, name="dollar_cost")
m.addConstr(14 * x1 + 14 * x2 >= 56, name="organization_score")
m.addConstr(13 * x1 + 18 * x2 >= 46, name="computer_competence")
m.addConstr(6 * x1 - 10 * x2 >= 0, name="six_x1_minus_ten_x2")
m.addConstr(18 * x1 + 7 * x2 <= 82, name="dollar_cost_upper")
qc1 = m.addQConstr(14 * x1**2 + 14 * x2**2 <= 158, name="organization_score_upper")
m.addConstr(13 * x1 + 18 * x2 <= 93, name="computer_competence_upper")
qc2 = m.addQConstr(19 * x1**2 + 2 * x2**2 <= 148, name="paperwork_upper")

m.optimize()

if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Paul: {x1.varValue}")
    print(f"Hours worked by Ringo: {x2.varValue}")
else:
    print("No optimal solution found.")
```