Using an iterative construction of the first-order intertwining technique,
we find k-parametric families of exactly solvable anharmonic oscillators wh
ose spectra consist of-a part isospectral to the oscillator plus k addition
al levels at arbitrary positions below E-0 = 1/2. It is seen that the 'natu
ral' ladder operators for these systems give place to polynomial nonlinear
algebras, and it is shown that these algebras can be linearized. The cohere
nt states construction is performed in the nonlinear and linearized cases.