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A characterization of the symmetry algebra of the nth order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra g determines - up to a point transformation - only one linear equation whose symmetry algebra is g and an algorithmic procedure is given to find the linearizing coordinates. The procedure is illustrated by several examples from literature.arXiv:1810.12334v2 fatcat:mzou2m74jbgefayshxukyx2cuy