  We consider a perturbed version of the second Painlev\'{e} equation
($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it
possesses solutions analogous to the celebrated Hastings-McLeod and
tritronqu\'ee solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type
solution of the perturbed equation is holomorphic, real-valued and positive on
the whole real-line, while the tritronqu\'ee-type solution is holomorphic in a
large sector of the complex plane. These properties also characterise the
corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising
because the perturbed equation does not possess additional distinctive
properties that characterise $\textrm{P}_{\textrm{II}}$, particularly the
Painlev\'e property.
