Where to find Single Phase Brushless DC Motor, or way of conversion

So how do you propose to reverse it? Simply reversing the supply voltage (i.e. by using an H-bridge) won't do it.

 

Although a BLDC motor is identical to a 3 phase PM motor it is reversed by changing the switching direction of the stator coils, of which only two of the three winding are switched at any one time.
IOW you require 6 switching devices not an H bridge, DC brushed motors use these method.
If you need a cheap motor to experiment there are BLDC RC motors, some are outrunners that revolve the outer rotor.
There is no such thing as a 1ph BLDC motor that I have come across, you require the phase angle provided by 3 stator coils.
M.

 

You change the rotation or direction of the energized stator fields.
M.

 

As Minder says, it needs 3phase ac. Perhaps a couple of interlocked changeover relays on 2 phases of the control module, which easiest would be say a hobbyking controller for maybe 10 or 15 bucks.

 

You need to build a driver, there are some dedicated IC's for this made originally by Motorola MC33030 etc and a few others.
You either need hall effect or sense current on power up and do a slight exercise of the motor to find the position of rotor.
If you go the RC route, there are a few DIY driver circuits out there. but you would have to modify to enable reverse mode.
You require all 3 conductors, for the 3 windings switching, the sequence for FWD would be 1&2 2&3 3&1 etc, for rev it would be 1&3 3&2 2&1 for e.g.
M.