Would a graph with such maximum weighted matchings exist? Edit Tony's answer is quite nice, but I meant something else. Sorry for editing again, I meant edges.
I am looking for a graph with 3 distinguished edges $xx'$,$yy'$,$zz'$ where $\deg(x)=\deg(y)=\deg(z)=1$.
One can chose arbitrary weights for the edges and the graph must satisfy:


*

*Must have at least two maximum weighted matchings in one of which all of the 3 distinguished edges are present and in the other all are not present.

*For all maximum weighted matchings (if more than 2) the distinguished edges are 
either all present or all not present.


Need this for a graph gadget and suspect it is quite unlikely to exist.
For only 2 distinguished edges a trivial solution is the path with 3 edges $v v' v'' v'''$.
 A: Such a gadget does not exist.
Proof for original vertex version. Suppose such a graph $G$ exists.  Let $x,y$, and $z$ be the distinguished vertices.  Let $M_1$ be a maximum weight matching which covers $x,y,z$, and let $M_2$ be a maximum weight matching which avoids $x,y,z$.  Suppose the edges of $M_1$ of $M_2$ are coloured red and blue respectively.  Consider $M_1 \triangle M_2$.  Every component of $M_1 \triangle M_2$ is either a path or an even cycle.  Since each of $x,y,z$ is covered by $M_1$ but not by $M_2$, $x,y,z$ are endpoints of path components of $M_1 \triangle M_2$.  There must exist a component of $M_1 \triangle M_2$ which contains exactly one of $x,y,z$.  Switching red and blue edges along this path produces another maximum weight matching which violates condition (2).  
Note that this proof does not actually assume that $x,y,z$ are of degree 1.   
Proof for edited edge version. Suppose such a graph $G$ exists.  Let $x,y$, and $z$ be the distinguished edges.  Let $M_1$ be a maximum weight matching which contains $x,y,z$, and let $M_2$ be a maximum weight matching which is disjoint from $x,y,z$.  Suppose the edges of $M_1$ of $M_2$ are coloured red and blue respectively.  Consider $M_1 \triangle M_2$.  Every component of $M_1 \triangle M_2$ is either a path or an even cycle.  Since each of $x,y,z$ is adjacent to a degree 1 vertex, each of $x,y$ and $z$ must be end edges of path components of $M_1 \triangle M_2$.  There must exist a component of $M_1 \triangle M_2$ which contains exactly one of $x,y,z$.  Switching red and blue edges along this path produces another maximum weight matching which violates condition (2).  
