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We analyze the equation of motion for a particle in the double -
well potential. We find the symmetries through Lie's method of group
analysis. In the corresponding quantum mechanical case, the method
of spectrum - generating $su(1, 1)$ algebra is used to find energy
levels as solutions of the Schordinger equation with double - well
potential, without solving the equation explicitly. Finally, we
discuss the symmetry version of double - well potential with the
vector field formalism}
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