  In the absence of any new physics signals at the Large Hadron Collider (LHC),
anomalous results at low energy experiments have become the subject of
increased attention and scrutiny. We focus on three such results from the LSND,
MiniBooNE (MB), and ATOMKI experiments. A 17 MeV pseudoscalar mediator ($a'$)
can account for the excess events seen in $^8$Be and $^4$He pair creation
transitions in ATOMKI. We incorporate this mediator in a gauge invariant
extension of the Standard Model (SM) with a second Higgs doublet and three
singlet (seesaw) neutrinos ($N_i, i=1,2,3$). $N_{1,2}$ participate in an
interaction in MB and LSND which, with $a'$ as mediator, leads to the
production of $e^+ e^-$ pairs. The $N_i$ also lead to mass-squared differences
for SM neutrinos in agreement with global oscillation data. We first show that
such a model offers a clean and natural joint solution to the MB and LSND
excesses. We then examine the possibility of a common solution to all three
anomalies. Using the values of the couplings to the quarks and electrons which
are required to explain pair creation nuclear transition data for $^8$Be and
$^4$He in ATOMKI, we show that these values lead to excellent fits for MB and
LSND data as well, allowing for a common solution. We obtain a representative
solution space for this, in the context of an important constraint that comes
from the decay $K^+ \rightarrow \pi^+ a' \, (a'\rightarrow e^+e^-$). We also
discuss other constraints on the model from both collider and non-collider
experiments as well as those from electroweak precision data, stability and
unitarity. We compute the contributions to the electron and muon $g-2$ up to
two loops for our model and discuss the results in the context of the current
theoretical and empirical scenario vis a vis these parameters. Finally, we
discuss future tests of the model in upcoming experiments.
