The problem is to analyze the equation $x^3 + y^3 = 3y$. We are asked to solve this equation. However, since no specific instructions are given (e.g., solve for $y$ in terms of $x$, or find integer solutions), we are simply rewriting the equation in different forms. It appears we're supposed to find the solutions to this equation. Without further information, finding all solutions can be complex. However, without additional instructions, let's rearrange this equation.

1. Problem Description

The problem is to analyze the equation $x^3 + y^3 = 3y$. We are asked to solve this equation. However, since no specific instructions are given (e.g., solve for $y$ in terms of $x$, or find integer solutions), we are simply rewriting the equation in different forms. It appears we're supposed to find the solutions to this equation. Without further information, finding all solutions can be complex. However, without additional instructions, let's rearrange this equation.

2. Solution Steps

We can rearrange the given equation as follows:

$x^3 + y^3 = 3y$

$x^3 + y^3 - 3y = 0$

This equation relates $x$ and $y$. Without additional context or constraints, we cannot obtain specific numerical solutions. We could express $x$ in terms of $y$ or $y$ in terms of $x$, but that's not explicitly requested. Let us solve for x:

$x^3 = 3y - y^3$

$x = \sqrt[3]{3y - y^3}$

Alternatively, let us see if we can factor $x^3 + y^3 - 3y = 0$. Factoring this cubic equation is difficult without knowing further constraints or having further instruction.

3. Final Answer

The equation can be rewritten as $x^3 + y^3 - 3y = 0$ or $x = \sqrt[3]{3y - y^3}$.

Final Answer: $x = \sqrt[3]{3y - y^3}$