Identical twins and heredity

The question came up again on an online forum. And, once again, folks hopped into the discussion with both feet, readily giving incomplete and misleading answers with confidence.
The question?
Can the tendency to have identical twins be inherited?
The most common answer and the prevailing theory is "no."
Most OBs will tell their patients that identical twins are fluke, an accident of nature, and that their chances of having another set are no greater than any other woman's chance of having identicals in the first place.
But how does that explain my neighbor's daughter and her family? She has three sets of identical twin boys who come trick-or-treating to our house every year.
And what about the woman whose daughter took dance lessons with my daughter? She is an identical twin and she has identical twin boys of her own.
Then there is the woman I met at a local bakery. She looked longingly at my boys (who were screaming at my attempt to get some coffee, unwilling to be pacified by cookies) and told me that her two daughters each have a set of identical twin girls.
It just doesn't make sense.
The reality is that scientists have no idea why some women have identical twins and some don't. Evidence does exist that many sets are flukes. For instance, there is no history in my family or in my husband's of identical twins as far as we know.
My side boasts a set of twins and a set of triplets way back in the olden days, but they were fraternal. My mother-in-law remembers a set of triplets birthed by a distant relative, but they also were fraternal.
So our boys probably were an accident of nature, an awesome accident.
But in other families, the frequency is too great for simple coincidence.
The journalist in me demanded that I do some research.
This is what I found:
A 2007 study, led by Dr. Dianna Payne, a visiting research fellow at the Mio Fertility Clinic in Japan, shows that identical twins form just after conception when an embryo collapses and splits in two. She discovered this by photographing growing embryos every two minutes in a lab using special computer software.
Her evidence negates previously held theories that the egg splits after it leaves its shell immediately before implantation and that, therefore, identicals either shared a placenta or had individual placentas that grew close together. The predominance of the previous theory explains why my OB insisted that our boys were fraternal until they were DNA tested.
Our boys each had their own placentas, which grew on polar opposite side of my uterus.
They were born in January of 2007, just before Dr. Payne and her colleagues went public with their research.
Dr, Payne's discovery has opened new paths for research into the potential genetic impact on identical twins. A study is currently underway that proposes that a male enzyme is involved. Scientist already know that the enzyme causes the embryo to collapse, but they are unsure who secretes it or why.
The study, as far as I know, is not yet public.
But I found this post-- written by a graduate bioengineering student who is helping to conduct the study at an unnamed university--on a Yahoo forum (http://www.twinslist.org/idfaq.htm). She says that the study also suggests some women carry a gene that prevents the enzyme from splitting the egg, and that men who produce the enzyme do not produce it every time:

"Thus far, the research shows that the enzyme is directly responsible for causing the splitting of the chromosomes, which results in the division of the cytoplasm which results in two eggs! There are a few (about 1%) that have alluded us so far and have shows no sign of the enzyme despite the fact that twins resulted. Thus, we have concluded that identical twinning can also be a random event. But in about 99% of the people tested, the enzyme is apparently the culprit. So, 99% is a darn good yield! So, according to our research, it is not a small percentage but almost the entire percentage."

This part of the post explains why we know so little:

"And yes, this study has been repeated already, but very, very little money is put into this research, so most of it is done on our own time. The reason for this is there is little to no medical advances that can come out of the research, just info and most of the money on medical and genetic research is for improving the outcome of diseases, etc. So, that is why it may be a while before this research is completed. We have to get about 5% of the public who have twins to complete our research (not all will go through tests, some will just answer questionnaires) before we can say that this represents the entire population, so you can see that this will take a while!"

So that's it.
The answer is that there is no definitive answer because the people in medical community, or rather the folks who fund their research, are just not not all that interested.
For now, identical twins remain a mystery.