In triangle $\triangle ABC$ with median from $B$ not perpendicular to $AB$ nor $BC$, we call $X$ and $Y$ points on $AB$ and $BC$, which lie on the axis of the median from $B$. Find all such triangles, for which $A,C,X,Y$ lie on one circumrefference.
1. **Define the problem and given conditions:**
   - We have a triangle $\triangle ABC$.
   - The median from $B$ is not perpendicular to $AB$ nor $BC$.
   - Points $X$ and $Y$ lie on $AB$ and $BC$ respectively, and they lie on the axis of the median from $B$.
   - We need to find all such triangles for which $A, C, X, Y$ lie on one circumcircle.

2. **Understand the properties of the median and the axis:**
   - The median from $B$ to $AC$ is the line segment that connects $B$ to the midpoint $M$ of $AC$.
   - The axis of the median from $B$ is the perpendicular bisector of the median $BM$.

3. **Use the condition that $A, C, X, Y$ lie on one circumcircle:**
   - For $A, C, X, Y$ to lie on one circumcircle, the points must be concyclic.
   - This implies that the opposite angles formed by these points must be supplementary.

4. **Analyze the geometric configuration:**
   - Since $X$ and $Y$ lie on the axis of the median from $B$, they are equidistant from the midpoint $M$ of $AC$.
   - The axis of the median is the perpendicular bisector of $BM$, so $X$ and $Y$ are symmetric with respect to $M$.

5. **Consider the special cases:**
   - If the $B$-symmedian is perpendicular to $AC$, then $B$ lies on the altitude from $A$ to $BC$.
   - This implies that $\triangle ABC$ could be isosceles with $AB = BC$ or right-angled at $B$.

6. **Verify the conditions for $A, C, X, Y$ to be concyclic:**
   - If $\triangle ABC$ is isosceles with $AB = BC$, then $X$ and $Y$ are symmetric with respect to $M$, and $A, C, X, Y$ can lie on a circle.
   - If $\triangle ABC$ is right-angled at $B$, then $A, C, X, Y$ can also lie on a circle because the right angle ensures that $A$ and $C$ are endpoints of the diameter of the circumcircle.

7. **Conclude the types of triangles:**
   - The triangles that satisfy the given conditions are either $B$-isosceles or $B$-right-angled.

\[
\boxed{\text{The triangles are either } B\text{-isosceles or } B\text{-right-angled.}}
\]