To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's define:
- $x_1$ as the number of laptops sold,
- $x_2$ as the number of desktops sold.

The objective function aims to maximize profit. Given that the profit per laptop is $210 and per desktop is $160, we can represent the total profit as:
\[ \text{Maximize:} \quad 210x_1 + 160x_2 \]

Now, let's list out the constraints based on the problem description:
1. The store can spend at most $20,000 on computers, with each laptop costing $500 and each desktop costing $400.
\[ 500x_1 + 400x_2 \leq 20000 \]
2. At least 20 laptops but at most 35 are sold each month.
\[ 20 \leq x_1 \leq 35 \]
3. The number of desktops sold is at most a third of the number of laptops sold.
\[ x_2 \leq \frac{1}{3}x_1 \]

All variables $x_1$ and $x_2$ must be non-negative integers since they represent quantities of items.

Thus, in symbolic notation with natural language objects:
```json
{
  'sym_variables': [('x1', 'laptops'), ('x2', 'desktops')],
  'objective_function': '210*x1 + 160*x2',
  'constraints': [
    '500*x1 + 400*x2 <= 20000',
    '20 <= x1 <= 35',
    'x2 <= (1/3)*x1',
    'x1 >= 0', 
    'x2 >= 0'
  ]
}
```

Here is the Gurobi code to solve this optimization problem:
```python
from gurobipy import *

# Create a new model
m = Model("Computer_Store_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="laptops")
x2 = m.addVar(vtype=GRB.INTEGER, name="desktops")

# Set objective function
m.setObjective(210*x1 + 160*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(500*x1 + 400*x2 <= 20000, "budget_constraint")
m.addConstr(x1 >= 20, "min_laptops")
m.addConstr(x1 <= 35, "max_laptops")
m.addConstr(x2 <= (1/3)*x1, "desktops_vs_laptops")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Laptops to sell: {x1.x}")
    print(f"Desktops to sell: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```