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Solving a system of equations using RowReduce?

I’d like to determine the value of $\alpha$ that minimizes $g(x, y, \alpha) = -x^3 + 2x^2y – 3xy^2 + y^3$.
Wolfram alpha is unable to solve this equation but I know that the equations simplify to $x = 3y$ and $y = \frac{1}{3}x$ so I tried to use a combination of RowReduce and Solve. My code is as follows:
RowReduce[{x – y – 3*y – 3*x + y – x == 0,
x – 3*y + x – 3*y – x + y == 0}, {x, y}, Integers]

I know this is a valid argument since it solves
RowReduce[{x + y + 3*y – 3*x + y – x – y + y – x == 0,
x + 3*y – x + y – x – y + y – x + y == 0}, {x, y}, Integers]

However, the command returns $\alpha$ as being a complex number and not an integer. Any suggestions?
Thanks!

A:

RowReduce[{x – y – 3*y – 3*x + y – x – y + y – x == 0,
x – 3*y + x – 3*y – x + y == 0}, {x, y}, Integers] /.
ComplexInfinity -> 0

$\left( \begin{array}{cc} -1 & -3 \\ -1 & -1 \\ \end{array} \right)$

A:

A second attempt (and using the command-line solver)
eqn2 = {x – y – 3*y – 3*x + y – x – y + y – x == 0,
x – 3*y + x – 3*y – x + y == 0};

s = Solve[eqn2, x] // ToRules // Evaluate

(* {{y -> –

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