  The expansion of ionization fronts in uniform and spherically symmetric
power-law density distributions is a well-studied topic. However, in many
situations, such as a star formed at the edge of a molecular cloud core, an
offset power-law density distribution would be more appropriate. In this paper
a few of the main issues of the formation and expansion of HII regions in such
media are outlined and results are presented for the particular cases where the
underlying power laws are 1/r^2 and 1/r^3. A simple criterion is developed for
determining whether the initial photoionized region will be unbounded, which
depends on the power-law exponent and the ratio of the equivalent Stroemgren
radius produced by the star in a uniform medium to the stellar offset distance.
In the expansion stage, the ionized volumes will eventually become unbounded
unless pressure balance with the external medium is reached before the
ionization front velocity becomes supersonic with respect to the ionized gas.
