We are given the system of equations:

1. $3x^2 + y^2 + z^2 = 1$
2. $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$

We are tasked with finding non-zero real values of $x$, $y$, and $z$ that satisfy both equations.

Step 1: Analyzing the first equation

The first equation is:

$3x^2 + y^2 + z^2 = 1$

This suggests that $x^2$, $y^2$, and $z^2$ are bounded such that their weighted sum equals 1, where $x^2$ is multiplied by 3.

Step 2: Analyzing the second equation

The second equation is:

$x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$

The left-hand side represents a sum of products of squared variables, while the right-hand side involves the cube of a sum of the variables, scaled by the product $xyz$. This equation is highly non-linear, making it challenging to solve directly.

Step 3: Trial values or symmetries

To proceed, one strategy is to try specific values for $x$, $y$, and $z$ that might satisfy both equations. Since the system is symmetric, we can explore solutions where $x = y = z$, and test whether this yields a valid solution.

Let’s assume $x = y = z = t$, then both equations simplify significantly.

For the first equation:

$3t^2 + t^2 + t^2 = 1 \quad \Rightarrow \quad 5t^2 = 1 \quad \Rightarrow \quad t^2 = \frac{1}{5} \quad \Rightarrow \quad t = \pm \frac{1}{\sqrt{5}}$

For the second equation:

If $x = y = z = t$, then: $t^4 + t^4 + t^4 = t^3(3t)^3 \quad \Rightarrow \quad 3t^4 = 27t^4$ This implies $3 = 27$, which is a contradiction.

Step 4: Conclusion

The assumption $x = y = z$ does not yield a valid solution. Therefore, other approaches, such as numerical methods or further exploration of asymmetry in $x$, $y$, and $z$, would be needed to solve this system.

Would you like further exploration of alternative methods, or any specific clarifications?

Follow-up questions:

1. Would you like to attempt different assumptions for $x, y, z$?
2. Do you want to explore graphical or numerical approaches to solving this system?
3. Shall we explore the possibility of complex solutions for the variables?
4. Would it be helpful to break down the meaning of the second equation in greater detail?
5. Do you need a geometric interpretation of the system?

Tip:

When dealing with non-linear systems, sometimes simplifying assumptions or substitutions can reduce complexity, but be mindful of contradictions.