We find under some mild assumptions that the most general potential of 1-dimensional conformal systems with time independent couplings is expressed as V = V 0 + V 1 , where V 0 is a homogeneous function with respect to a homothetic motion in configuration space and V 1 is determined from an equation with source a homothetic potential. Such systems admit at most an SL(2, R) conformal symmetry which, depending on the couplings, is embedded in Diff(R) in three different ways. In one case, SL(2, R) is also embedded in Diff(S 1 ). Examples of such models include those with potential V = αx 2 +βx −2 for arbitrary couplings α and β, the Calogero models with harmonic oscillator couplings and non-linear models with suitable metrics and potentials. In addition, we give the conditions on the couplings for a class of gauge theories to admit a SL(2, R) conformal symmetry. We present examples of such systems with general gauge groups and global symmetries that include the isometries of AdS 2 × S 3 and AdS 2 × S 3 × S 3 which arise as backgrounds in AdS 2 /CF T 1 .