does it exist examples of partialsplicing of mRNA where at least one
intron is conserved

The immediate regions of introns that surround the exons contain
information that says to the RNA Polymerase that " the exons end here
and stop doing what you are doing ! ". In the theoritical utopic
world, there should be NO mRNA containing any intron components. If
something wrong happens, those mRNAs will be checked in the long
processes before it leaves the nucleus, and marked for degradation.

To put it simple. if the widely accepted central-dogma is correct,
then there should be virtually no introns in the mRNA of any organism
!!

Apologies to um-ga, but after an RNA is transcribed, not all of the
potential introns are removed by splicing.  If a gene is broken into
five segments (A through E), then splicing can take on a lot of forms.
 Some patterns that have been observed are:

1 - conditional splicing (ABCDE becomes ABCDE (not splcied at all) or
ABDE)
2 - mutually-exclusive splicing (ABCDE may become ABCE or ABDE)
3 - shared splice start (ABCDE becomes ACDE or ADE)
4 - shared splcie end (ABCDE becomes ABCE or ABE)

Splicing may also take place between two different RNAs (called
trans-splicing), so that ABCDE and FGHIJ may splice with WXYZ to give
WXCDE and WXHIJ.

These alternative forms of splicing are not mistakes or errors in
biology.  The determining factor between male and female fruit flies
is an alternative splice.  Many neuroreceptors (involved in neuron
function) or muscle proteins are alternatively spliced, and each
product has a different function.  Alternative splicing is a very
complicated.  You can search for Alternative Splicing and find a lot
of college class notes (the first I found was
http://www.blc.arizona.edu/marty/411/Modules/altsplice.html), and see
how much there is.