Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.
To decipher the given equality, we need to analyze the constraints and the possible values for the digits. The equality is:

\[
(\overline{LARN} - \overline{ACA}) : (\overline{CYP} + \overline{RUS}) = C^{Y^P} \cdot R^{U^S}
\]

where different symbols correspond to different digits, and equal symbols correspond to equal digits. All digits are different from 0.

1. **Determine the range of values for $\overline{LARN} - \overline{ACA}$:**
   - The maximum value for a 4-digit number $\overline{LARN}$ is 9876.
   - The minimum value for a 3-digit number $\overline{ACA}$ is 121 (since digits are non-zero and different).
   - Therefore, the maximum value for $\overline{LARN} - \overline{ACA}$ is:
     \[
     \overline{LARN} - \overline{ACA} \leq 9876 - 121 = 9755
     \]

2. **Determine the range of values for $\overline{CYP} + \overline{RUS}$:**
   - The minimum value for a 3-digit number $\overline{CYP}$ is 123.
   - The minimum value for a 3-digit number $\overline{RUS}$ is 256.
   - Therefore, the minimum value for $\overline{CYP} + \overline{RUS}$ is:
     \[
     \overline{CYP} + \overline{RUS} \geq 123 + 256 = 379
     \]

3. **Analyze the left-hand side (LHS) of the equality:**
   - Given the ranges, the LHS of the equality is:
     \[
     \frac{\overline{LARN} - \overline{ACA}}{\overline{CYP} + \overline{RUS}} < \frac{9755}{379} \approx 25.74
     \]
   - Therefore, the LHS is less than 26.

4. **Analyze the right-hand side (RHS) of the equality:**
   - The RHS is given by \(C^{Y^P} \cdot R^{U^S}\).
   - Since all digits are different and non-zero, and one of the sets \(\{C, Y, P\}\) or \(\{R, U, S\}\) does not contain the digit 1, we need to consider the minimum values for \(a^{b^c}\) where \(a, b, c \geq 2\).
   - The smallest value for \(a^{b^c}\) with \(a, b, c \geq 2\) is \(2^{2^2} = 2^4 = 16\).
   - However, since the digits are different and non-zero, the smallest possible value for \(C^{Y^P} \cdot R^{U^S}\) is greater than 25.

5. **Conclusion:**
   - Since the LHS is less than 26 and the RHS is greater than 25, it is impossible for the equality to hold true under the given constraints.

\[
\boxed{\text{No solution}}
\]