We all understand what is meant by reincarnation, and in this context it is well established as a fact; even psychologists have proven that we have past lives (but this information is not allowed to become established and brought to the attention of the public).
Almost the entirety of the New-Age field acknowledges the validity of reincarnation and also Eastern philosophies, contrary to Christianity and science. There are, however, one or two other sources which could be placed in the New-Age category which do not agree with reincarnation.
The Urantia Book---a perfectly written work of 2000 pages by over a hundred spiritual beings (apparently not channelled but appeared in the form of papers in 1934)---emphatically denies reincarnation as 'stultifying'. If we assume this is a true statement, can we find any way of reconciling this viewpoint?
The book does give an afterlife description in which graded levels have been created for man's gradual evolution from the one basic physical life through over 600 bodies of decreasing density and increasing frequency into higher and higher spiritual worlds. This could be a satisfactory explanation except that it does not state that it is an alternative route--it actually appears to deny reincarnation.
There is also the channelled book The Book of James, from the philosopher William James. This is in the New-Age/spiritualist category but again denies positively the validity of past lives. It adds, however, that a person on rare occasions may return after physical death from an Earth bound condition and take on another body. But states that this is unfortunate and not conducive to the evolution of the individual. We can agree with this since it is not the proper method of reincarnation in which the being goes to the astral planes or above returning to the higher-self with great benefit before eventually setting up another incarnation.
We can of course say that there is really no past lives since everything is basically simultaneous. But if the above sources meant this they would surely state this. There is still another possible way out. It would be correct to say that we do or do not have past lives depending on the context. In the context of the lower-self personality we do not have more than one life, but in the context of the higher-self we have many lives of different personalities, though projected from the same spiritual being. Again if this was the explanation of the discrepancies one would expect them to be stated. We are thus left with a puzzle as to why these books create this confusion (assuming they are incorrect).
Since writing the above it has become apparent that Urantia is in fact a realm of 'advanced' fallen angelics who are 'recruiting'---to put it mildly---humans. Thus any statement made by them can be regarded as suspect.