Identify a non-Western culture that still has strong ties to its own traditional culture but integrates with current Western...

I would say that Japan is a culture that fits this
description.  It has thoroughly Westernized in many ways, but it has not done so
completely.

We all know that Japan has become quite
Westernized.  Japan's economy was until just recently the second largest in the world. 
It built that economic power on industrial power, which is a very Western thing
historically.  It has become a democratic country, also a Western
thing.

At the same time, Japan has not abandoned its
traditional culture.  Though it is not as prevalent as before, Japanese are (of course
this is not true of all Japanese) still very much more reserved and formal than most
Westerners are.  Japanese are still very much tied to their traditional foods.  They
still do very quintessentially Japanese things like cherry blossom-viewing and autumn
leaf-viewing.  In these ways, Japan is still quite distinctively Japanese even as it has
become an industrial, capitalist democracy.