We find under some mild assumptions that the most general potential of one-dimensional conformal systems with time-independent couplings is expressed as V = V0 + V1, where V0 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 conformal symmetry which, depending on the couplings, is embedded in in three different ways. In one case, is also embedded in Diff(S1). Examples of such models include those with potential V = αx2 + βx-2 for arbitrary couplings α and β, the Calogero models with harmonic oscillator couplings and nonlinear models with suitable metrics and potentials. In addition, we give the conditions on the couplings for a class of gauge theories to admit a conformal symmetry. We present examples of such systems with general gauge groups and global symmetries that include the isometries of AdS2 × S 3 and AdS2 × S3 × S3 which arise as backgrounds in AdS2/CFT1.