Given an electromagnetic field in a frame of reference, find all other frames of reference where the electric and magnetic field are parallel

This is actually an exercise in Landau-Lifshitz's book. Their solution goes as follows.

After we have found a frame of reference where $$\mathbf E$$ and $$\mathbf B$$ are parallel (let's call the common direction $$\mathbf n$$), every other frame of reference obtained by boosting along $$\mathbf n$$ is such a frame of reference. So they proceed to find the frame of reference where the electric and magnetic field are parallel, searching among those obtained by boosting along $$\mathbf E\wedge \mathbf B$$. In order to do so, they impose that the cross product between the new electric and magnetic fields is zero, and find a condition for the relative speed between the given frame of reference and the new one.

My problem is, are they sure a priori to find a solution where the boost is perpendicular to both $$\mathbf E$$ and $$\mathbf B$$? Also, if the original $$\mathbf E$$ and $$\mathbf B$$ are not parallel, is there any other frame of reference where they are parallel, other than those that they found (ie, the one boosted perpendicularly, and all the ones boosted from the latter along the common direction)?

I hope I was clear enough in explaining the problem. If anything needs to be explained better, please comment, I will edit the question. Thanks in advance!

• A good reference here. – Ng Chung Tak Oct 12 '18 at 10:26
• 1. I don't understand how the "initial boost can be sought perpendicular to both E and B". I would say it should be sought perpendicular to the common direction after the initial boost. 2. Even then, isn't this assuming the total boost is along a direction which is given by the vector sum of the two velocities? – renyhp Oct 12 '18 at 11:32
• In the reference given by @NgChungTak, and also in [chegg.com/homework-help/…, it is shown how to find a boost in the direction perpendicular to both E and B that makes E and B parallel. Unfortunately neither reference explains how to take the next step and find all initial boosts that make E and B parallel. Turns out my question [physics.stackexchange.com/questions/460253/… is essentially the same as yours. – S. McGrew Feb 12 at 14:53
• Maybe the extra clues mentioned in my question [physics.stackexchange.com/questions/460253/… will be helpful. If you find the answer, please post it to my question too. – S. McGrew Feb 12 at 14:56