This puzzle replaces all numbers with other symbols.

Your job, as the title suggests, is to find what number fits in the place of $\bigstar$.

All symbols abide to the following rules:

 1. Each symbol represents integers and only integers. This means fractions and irrational numbers like $\sqrt2$ are not allowed. However, negative numbers and zero are allowed.
 2. Each symbol represents a __unique__ number. This means that for any two symbols $\alpha$ and $\beta$ which are in the same puzzle, $\alpha\neq\beta$.
 3. The following equations are satisfied (this is the heart of the puzzle):
$$
\text{I. }\alpha^\beta=\beta^\alpha
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\text{II. }\alpha+\beta=\gamma
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\text{III. }\delta+\delta=\gamma
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\text{IV. }\gamma+\delta=\varepsilon
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\text{V. }\delta^\beta=\varepsilon^\alpha
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\text{VI. }\zeta=\delta\times\delta+\eta
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\text{VII. }\theta=\alpha\times\gamma-\eta
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\text{IIX. }\bigstar=\zeta+\theta
$$

###What is a Solution?###
A solution is an integer value for $\bigstar$, such that, for the group of symbols in the puzzle $S_1$ there exists a one-to-one function $f:S_1\to\Bbb Z$ which, after replacing all provided symbols using this function, satisfies all given equations.

###What is a Correct Answer?###
An answer is considered correct if you can prove that a certain value for $\bigstar$ is a solution. This can be done easily by getting a function from every symbol in the puzzle to the correct integers (that is, find an example for $f:S_1\to\Bbb Z$).

An answer will be __accepted__ if it is the first correct answer to also prove that the solution is the ___only___ solution. In other words, there is no other possible value for $\bigstar$.

Good luck!

Previous puzzles in the series:

 [Puzzle 1](https://puzzling.stackexchange.com/questions/59396/find-the-value-of-bigstar-puzzle-1)