I wish to hang a ceiling fixture where there are 2-two wire cables present. When I disconnect the wires it turns off power to the existing ceiling light fixtures which are connected to a 3 way switch. These wires have a constant power to them. How can I utilize these cables?

2 2 wire cables that turn off light when disconnected are probably travellers, Depending on the position of one of the 3 ways(1 wire will be hot or the other), Neither constant, and there is no neutral from what you describe.

Thanks for the answer.Now how do I connect the new ceiling fixture to the above wires so that this new fixture can be turned on & off like the existing fixtures

I believe what you are describing is just two fixtures controlled by a set of 3-way switches. The power (and the neutral) come from the switch to the first fixture and then go on to the second fixture. Just connect the two black wires and the black of the light together, connect the two white wires and the white of the light together. If you don't know the off setting of the switches , turn the power to the circuit off at the breaker panel.

You may be able to hook on to the 2 connections as hk mentiones, if that is the neutral and switchleg, and not the travellers, connect fixture, see what happens, no sparks will fly, it will work or it won't.

I did hook up the fixture as mentioned above... all blacks together & all whites together... but this gave me constant light/power to the new fixture and I was unable to turn off & on this fixture... How do I control this new fixture?

Lets start over.

a. You are replacing a light which is one of two lights on 3-way switches.
When you say that there is 2 2-wire cables , you mean in the box where you are
Replacing the light.
b. You are adding a second light to to an existing light which is controlled by 3-way
Switches. When you say there are 2 2-wire cables you mean in the box of the first
Or original light.

Tell us a little more about what you are doing, and what you have done. Tell us what wires you have in the other fixture box.

First off, how was the old fixture connected? Or is it that you don't remember?
This would be a big help obviously.

Second, the wires in the ceiling have NOTHING to do with the 3-way switching.
I would have assumed you had a power feed and a switch loop, but connecting black to black and white to white would have created a direct short in that case.
When you did connect it that way did you try the switches, both of them? If the switch was off then the short would have been avoided.

As opposed to guessing and telling you to try every different combination, which is VERY unwise and can be very unsafe, you'll need to tell us EXACTLY what wires and what colors, and how they are connected, in both switch boxes.
DO NOT take anything else apart or un-do anymore splices!

If it light stays on no matter what switching, it MAY have been meant for pull chain switching? Smoke or heat detector?