Let $AB$ be a chord on parabola $y=x^2$ and $AB||Ox$. For each point C on parabola different from $A$ and $B$ we are taking point $C_1$ lying on the circumcircle of $\triangle ABC$ such that $CC_1||Oy$. Find a locus of  points $C_1$.
1. Let the coordinates of points \( A, B, C, C_1 \) be \( (a, a^2), (-a, a^2), (c, c^2), (c, d) \) respectively. Since \( AB \parallel Ox \), the y-coordinates of \( A \) and \( B \) are the same.

2. The center \( O(x_0, y_0) \) of the circumcircle of \( \triangle ABC \) must satisfy the equation of the circle:
   \[
   (x - x_0)^2 + (y - y_0)^2 = r^2
   \]

3. For points \( A \) and \( B \), we have:
   \[
   (a - x_0)^2 + (a^2 - y_0)^2 = (-a - x_0)^2 + (a^2 - y_0)^2
   \]
   Simplifying this, we get:
   \[
   (a - x_0)^2 = (-a - x_0)^2
   \]
   \[
   a^2 - 2ax_0 + x_0^2 = a^2 + 2ax_0 + x_0^2
   \]
   \[
   -2ax_0 = 2ax_0
   \]
   \[
   x_0 = 0
   \]

4. For points \( A \) and \( C \), we have:
   \[
   a^2 + (a^2 - y_0)^2 = c^2 + (c^2 - y_0)^2
   \]
   Simplifying this, we get:
   \[
   a^2 + a^4 - 2a^2y_0 + y_0^2 = c^2 + c^4 - 2c^2y_0 + y_0^2
   \]
   \[
   a^2 + a^4 - 2a^2y_0 = c^2 + c^4 - 2c^2y_0
   \]
   \[
   a^2 - c^2 + a^4 - c^4 = 2y_0(a^2 - c^2)
   \]
   \[
   (a^2 - c^2)(1 + a^2 + c^2) = 2y_0(a^2 - c^2)
   \]
   Since \( a \neq c \), we can divide both sides by \( a^2 - c^2 \):
   \[
   1 + a^2 + c^2 = 2y_0
   \]
   \[
   y_0 = \frac{1 + a^2 + c^2}{2}
   \]

5. For points \( C \) and \( C_1 \), we have:
   \[
   c^2 + (c^2 - y_0)^2 = c^2 + (d - y_0)^2
   \]
   Simplifying this, we get:
   \[
   (c^2 - y_0)^2 = (d - y_0)^2
   \]
   \[
   c^4 - 2c^2y_0 + y_0^2 = d^2 - 2dy_0 + y_0^2
   \]
   \[
   c^4 - 2c^2y_0 = d^2 - 2dy_0
   \]
   \[
   c^2(c^2 - 2y_0) = d(d - 2y_0)
   \]
   Since \( d \neq c^2 \), we can solve for \( d \):
   \[
   d = 2y_0 - c^2
   \]
   Substituting \( y_0 = \frac{1 + a^2 + c^2}{2} \):
   \[
   d = 2 \left( \frac{1 + a^2 + c^2}{2} \right) - c^2
   \]
   \[
   d = 1 + a^2
   \]

Thus, the locus of points \( C_1 \) is the line \( y = 1 + a^2 \), where \( a \) is the x-coordinate of point \( A \).

The final answer is \( \boxed{ y = 1 + a^2 } \).