Question: polynomial include parameter

I have a problem in determine the relation between two variables in a polynomial equation f(l2,v)=0. The results of two methods puzzled me.
The first method I tried is use the fsolve command.
In equation f(l2,v)=0,the range for v is [3405,5054] and l2 is a complex variable.
I plotted the figure for the l2=l2(v) in complex plane.Because the degree of l2 is 8, I have eight curves in the figure.
Obviously, the order in the result of fsolve is indefinite, so it can be seen that even in the same branch,the color of l2 is different.
Secondly, I use the dsolve command to solve the DE diff(l2(v),v)=subs(l2=l2(v),-diff(f(l2,v),v)/diff(f(l2,v),l2)), use the evalc to split the complex DE, use the fsolve to provide an initial value and then plot the 2nd figure.
After that, I got two problems.
1.There are some points which exist in the 1st figure didn't appear in the 2nd figure.
2.The points on the 2nd figure depend on the initial value provided by the fsolve command, which means instability.
I just want get the eight functions l2[i]=l2[i](v),i=1..8, which means eight branches for l2 in complex plane.

Thank you very much for any advice.the mw file is below.

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