fixed income - How to adjust butterfly 2s5s10s swaps trade for directionality?

I am looking into a 2s5s10s swaps idea using a 50-50 weighting scheme, where it's 2 times the 5 year minus the 2 year and 10 year. However, there is a correlation between the butterfly spread and the slope of the curve (2s10s) and also with the 5 year. I want to adjust the weights of my butterfly to remove the directionality such that I still want the body to have a weight of 2.

Essentially, I think it should be X : 2 : Y but I am having a hard time.

Answer

I think he's saying that if

$$Fly=b_1(2s-10s) + b_2(5s) + error$$

But

$$Fly= 2s-2(5s)+10s$$

Then doing some algebra ,

$$(1-b_1)2s -(2-b_2)5s + (1+b_1)10s = error$$

Hence the weights of the fly that give pure noise , un correlated to curve and rate , are as given above. If you want 2 in the middle , you have to rescale.