The question is expressed in the title. Directly using ordinary least square fitting is not a good idea, because the intercepts are different among these lines, and a single linear equation is unreasonable. Is there any good mathematical method to solve this problem?
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Projection pursuit might be an option ; it should reveal the direction orthogonal to the clusters
https://www.stat.washington.edu/wxs/Visualization-papers/projection-pursuit.pdf
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If the clusters are distinguished by some characteristic variable (GROUP) you can use software such as SAS or similar to do the regressions BY GROUP. (This will generate the slope and intercept for each group.) If you have a different problem such as simultaneously determine the groups to use and the slope for those group, you might want to try some kind of cluster analysis to determine group membership first and then do the regressions.
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