Hi,

ok so a break in any line represents a change or transformation,  in your case it is the heart line which relates to your emotions.  Therefore, there is a change taking place with your emotions, somehow someway or the other, you are the only one who probably knows exactly in what way that is.  I know quite a few people with breaks in their heart lines, and there emotions are very easily changed and stirred, one minute they will be nice towards you, the next they will be verbally fighting with you and will snap at you.  Therefore, because you have a broken heart line doesn't literally mean you are broken hearted although a lot of times it is the case.

Also, your fingers are covered with very fine lines which can be seen at the top of the picture.  These are indicators of stress or a worrisome nature, which is corroborated by the network of lines on your hand.  Therefore, in your case from what i have heard i would say it may not be so much actual stress but more of a worrisome disposition that results in the network of lines.

The sun line or line of Apollo is the line under the ring finger.  As far as exploiting the potential of your sun line goes, i would say that the expression of the sun line is being interfered by the network of lines on the hand which aren't allowing it to express itself.  I would say you need to learn to be more relaxed and more open.  There are many ways of doing this, a hobby could of course be one way.  As far as developing the line of Apollo, studying palmistry could be one way to develop the line of Apollo after all he was the god of fortune telling among many other things.  

As far as accidents are concerned i would not worry about such things.  You should look at palmistry and other pseudosciences more as ways of understanding your energies and behavior patterns, how to modify them, and to gain an understanding of how certain actions will lead to certain results.  Your future is not a ball on a chain that you drag around with you, you have creative force in molding and shaping your future.  Therefore, i would not needlessly worry about an accident in life, but i would try to understand how to better use your energy to maximize your potential.  Therefore, if you would like to live a serene existence you have to go out and create such an environment through your thoughts and actions, someone saying so and so will happen has no significance or importance unless you actively work at it.

Hope i was helpful!