simulate this circuit – Schematic created using CircuitLab
Figure 1. Interpretation of OP's circuit with additional diode D2.
A PMDC motor M2 is connected to a constant regulated voltage source. To protect the regulator chip from the flyback voltage, a diode has been added.
That much is clear.
From the same unregulated voltage source another motor is also driven.
M1 is powered directly off the solar panel.
The voltage regulator has an inbuilt polarity protection and as such I don't want to add another diode in series with the circuit so as to have an unnecessary drop.
OK. No series diode added.
Also if a flyback diode is used across M1 motor, it will create a short under reverse polarity.
This isn't clear. How will reverse polarity occur? There are no switches anywhere in your circuit.
What will be the effect of flyback voltage generated across M1 motor on the regulator chip? If possible, please show the conventional current flow in the circuit.
Just add D2. It's normally reversed biased. If the solar panel is suddenly disconnected current will continue to flow in the motor and D2 until the motor stops. No problem.
Without D2 a negative voltage spike will be generated on disconnection of the power source. This may destroy the regulator.
If you need further clarification then please add a proper schematic to your question. There's a schematic button on the editor toolbar. Keep positive at the top, negative at the bottom and, in general, power flow from left to right.
simulate this circuit
Figure 2. Zero volt-drop reverse polarity protection.
The scheme of Figure 2 protects the circuit against reverse polarity connection of the solar panel or battery but has zero voltage drop.