  We introduce a system with competing self-focusing (SF) and self-defocusing
(SDF) terms, which have the same scaling dimension. In the one-dimensional (1D)
system, this setting is provided by a combination of the SF cubic term
multiplied by the delta-function, $\delta (x)$, and a spatially uniform SDF
quintic term. This system gives rise to the most general family of 1D-Townes
solitons, the entire family being unstable. However, it is completely
stabilized by a finite-width regularization of the $\delta $-function. The
results are produced by means of numerical and analytical methods. We also
consider the system with a symmetric pair of regularized $\delta $-functions,
which gives rise to a wealth of symmetric, antisymmetric, and asymmetric
solitons, linked by a bifurcation loop, that accounts for the breaking and
restoration of the symmetry. Soliton families in 2D versions of both the
single- and double-delta-functional systems are also studied. The 1D and 2D
settings may be realized for spatial solitons in optics, and in Bose-Einstein
condensates.
