Find all real solutions of the system of equations: 
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
1. **Case \( n = 1 \):**
   \[
   \sum_{k=1}^1 x_k^1 = a^1 \implies x_1 = a
   \]
   This is straightforward since there is only one equation and one variable.

2. **Case \( n \geq 4 \):**
   \[
   \sum_{k=1}^n x_k^i = a^i \quad \text{for} \quad i = 1, 2, \ldots, n
   \]
   For \( i = 4 \):
   \[
   a^4 = \left( \sum_{k=1}^n x_k^2 \right)^2 = \sum_{k=1}^n x_k^4 + 2 \sum_{1 \leq i < j \leq n} x_i^2 x_j^2
   \]
   Since \( \sum_{k=1}^n x_k^4 = a^4 \), it follows that:
   \[
   2 \sum_{1 \leq i < j \leq n} x_i^2 x_j^2 = 0
   \]
   This implies:
   \[
   \sum_{1 \leq i < j \leq n} x_i^2 x_j^2 = 0
   \]
   Therefore, at most one of the \( x_i \) can be nonzero, and it must equal \( a \).

3. **Case \( n = 3 \):**
   \[
   a^2 = x_1^2 + x_2^2 + x_3^2
   \]
   and
   \[
   a^2 = (x_1 + x_2 + x_3)^2
   \]
   Expanding the second equation:
   \[
   a^2 = x_1^2 + x_2^2 + x_3^2 + 2(x_1 x_2 + x_2 x_3 + x_3 x_1)
   \]
   Equating the two expressions for \( a^2 \):
   \[
   x_1^2 + x_2^2 + x_3^2 = x_1^2 + x_2^2 + x_3^2 + 2(x_1 x_2 + x_2 x_3 + x_3 x_1)
   \]
   This simplifies to:
   \[
   2(x_1 x_2 + x_2 x_3 + x_3 x_1) = 0 \implies x_1 x_2 + x_2 x_3 + x_3 x_1 = 0
   \]
   Using the identity:
   \[
   x_1^3 + x_2^3 + x_3^3 - 3x_1 x_2 x_3 = (x_1 + x_2 + x_3)(x_1^2 + x_2^2 + x_3^2 - x_1 x_2 - x_2 x_3 - x_3 x_1)
   \]
   Since \( x_1 + x_2 + x_3 = a \) and \( x_1^2 + x_2^2 + x_3^2 = a^2 \):
   \[
   a^3 - 3x_1 x_2 x_3 = a(a^2 - 0) \implies a^3 - 3x_1 x_2 x_3 = a^3 \implies x_1 x_2 x_3 = 0
   \]
   Therefore, at least one of \( x_1, x_2, x_3 \) must be zero. Without loss of generality, let \( x_3 = 0 \):
   \[
   x_1 x_2 = 0
   \]
   Thus, at most one of \( x_1 \) or \( x_2 \) is nonzero, and it must equal \( a \).

4. **Case \( n = 2 \):**
   \[
   a^2 = x_1^2 + x_2^2
   \]
   and
   \[
   a^2 = (x_1 + x_2)^2
   \]
   Expanding the second equation:
   \[
   a^2 = x_1^2 + x_2^2 + 2x_1 x_2
   \]
   Equating the two expressions for \( a^2 \):
   \[
   x_1^2 + x_2^2 = x_1^2 + x_2^2 + 2x_1 x_2
   \]
   This simplifies to:
   \[
   2x_1 x_2 = 0 \implies x_1 x_2 = 0
   \]
   Therefore, at most one of \( x_1 \) or \( x_2 \) is nonzero, and it must equal \( a \).

The final answer is \( \boxed{ x_i = a } \) for at most one \( i \) and \( x_j = 0 \) for all \( j \neq i \).