PhD supervisor is not willing to co-supervise? I started my PhD two years ago on a specific problem. The problem is to study a given computer science structure from one specific perspective. My current advisor is an expert in that perspective and I got some results and publications with him.

However, during the last 6 months, I started to worry about my PhD thesis as there are no more room for novel results in that direction (several people tackled it recently). There are still some room for minor results and my advisor urged me to tackle them for my PhD thesis. 

I am no longer interested in pursuing that direction and shifted towards the other perspectives of the problem as I believe there are still major results to come out of them. My advisor has no objection on pursuing these directions by myself (at the end he will put his name in whatever comes out of it!). Recently, I took a class with another professor and tackled one of the important directions of the problem as a class project and had incomplete but interesting results. The class professor was very happy with my work and advised me really well during the class semester to start tackling the problem. After couple of months of working, it seems I can really get something nice out of this.

I started to believe that, for my PhD thesis, the best thing is to have the class professor as a co-advisor. This will allow me to put in action and test rigorously several ideas I have in the other direction and make my PhD thesis more valuable. 

I spoke with my current supervisor about the co-supervision but he insisted to make it as a collaboration instead of co-supervision. He has good relation with the other professor. And if it matters, I have my own external funding with no financial obligation from the current supervisor.


  What should I do in such case?


EDIT on the collaborator vs co-advisor thing:
Because I had a specific problem and had a class with him (which makes it easy to work on the other direction) and the results were almost ready couple of months later, there was a collaboration. My advisor is not willing to collaborate again and urging me to do the minor results instead of pursuing the other direction. The reason I look for co-supervision is  


Consult him in many scientific results I am not good at (neither my current advisor).
No commitment in advance for collaborations/ papers.
co-advisor may suggest a new perspective to the problem instead of accepting/rejecting what me and my advisor suggest.
I think @BobBrown summarized everything that needs to be said.


  Brown's Rule for finishing the Ph.D.: Listen closely to what your
  supervisor tells you to do and do it. You can do the other research
  later.


In detail:


You have some publications from your cooperation with your main supervisor. Excellent.
Publish another paper with your new collaborator, if you have not done already. 
Then do another paper (those minor results) with your advisor. Finish everything there is to be done on this direction fast.
Finish PhD. 
Do whatever you want afterwards.


This is by far the easiest route to your PhD.