Identifying 4 coil 2 wire motor and run it from DC
As the label say, this is out of a Ricool-3 blower.
See catalog here pages 56 &amp; 57.

The original blower has the controller inbuilt and is speed controllable using PWM. However, it has a 3 x 10 30 pin connector with a host of uncertain or outright inscrutable functions. 

Do you have the complete original unit?  

If you do not have the original controller then a simple reversing polarity drive and suitable hall rotation sensor(s) will work - as described below. 



There is probably either an external rotation sensor or the driver IC that controlled the motor has back-EMF sensing.  If you have the original control electronics you can probably use them to make the job much easier.

If the motor has only 2 wires connecting the stator then it is a crude but clever BLDCM (brushless DC motor) implementation.  

When voltage of one polarity is applied the 4 coils have magnetic polarity of NSNS. When the applied voltage polarity is reversed the magnets will swap to SNSN. Initially a rotor permanent magnet above or near a pole of the same polarity N-N or S-S will be repelled. and a rotor magnet above or near a pole of opposite polarity S-N or N-S will be attracted. Once the rotor position has mechanically stabilised so that all rotor magnets are aligned with poles of opposite polarity, reversing the applied voltage will make the rotor move 1/4 turn to again cause stability. If, as the rotor just about reaches the new position, the applied voltage polarity is again reversed the rotor will "continue on its way" due to inertia, 'heading for' the next stable point which is now 1/4 turn away. If voltage reversals occur every 1/4 turn the rotor will "chase" the moving magnetic field.
As the chase continues the rotor will move faster and the time between 1/4 turns will decrease. The controller needs to "know" when the rotor is approaching the latest table position so that it can change voltage polarity.
This is achieved with a sensor or sensors - either external hall sensors or by sensing the induced "back EMF voltages). 

The existing controller will include the means of sensing. Using the controller if available may be the easiest approach. If it is an integral part of a microcontroller then this may not be possible. In that case, adding a Hall sensor to detect rotor position and switching coil polarity based on the result would probably be the easiest approach. A single sensor will not allow rotation direction to be controlled. Two hall sensors offset from each other by a fraction of 1/4 of a turn will allow direction to be detected and so controlled. eg If S1 (sensor 1) &amp; S2 outputs are low then whichever goes high 1st shows which way the rotor is turning. Simple logic can be used to change the drive signal if direction is incorrect.

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Test Circuit:

The L298 IC datasheet here is a dual half bridge driver IC that could be used to make a test circuit.

Using the diagram below, application of a 0V/5V square wave at Vin will produce a square wave toggling motor drive signal at Vout.
The IC needs 5V on V22 (pin 9 in diagram) and motor voltage on Vs (pin 4).
This circuit has NO feedback signal so will not drive the motor well but should show that it is working.  



The inverter is one portion of any 5V capable inverter 

Apply 5V to Vss, probably OK to try 5V to Vs initially, and apply a 0/5V square wave at say 100 Hz to Vin.
Turn on the supply - the motor will "jiggle" and may or may not turn. Spin the shaft by hand (with due care) and the motor will quite possible run at about one turn per square wave cycle. 

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