  Let $A_\pm>0$, $\beta\in(0,1)$, and let $Z^{(\alpha)}$ be a strictly
$\alpha$-stable L\'evy process with the jump measure $\nu(\mathrm{d}
z)=(C_+\mathbb{I}_{(0,\infty)}(z)+
C_-\mathbb{I}_{(-\infty,0)}(z))|z|^{-1-\alpha}\,\mathrm{d} z$, $\alpha\in
(1,2)$, $C_\pm\geq 0$, $C_++C_->0$. The selection problem for the model
stochastic differential equation $\mathrm{d} \bar
X^\varepsilon=(A_+\mathbb{I}_{[0,\infty)}(\bar X^\varepsilon) -
A_-\mathbb{I}_{(-\infty,0)}(\bar X^\varepsilon))|\bar X^\varepsilon|^\beta
\,\mathrm{d} t +\varepsilon \mathrm{d} Z^{(\alpha)}$ states that in the small
noise limit $\varepsilon\to 0$, solutions $\bar X^\varepsilon$ converge weakly
to the maximal or minimal solutions of the limiting non-Lipschitzian ordinary
differential equation $\mathrm{d} \bar x=(A_+\mathbb{I}_{[0,\infty)}(\bar x)-
A_-\mathbb{I}_{(\infty,0)}(\bar x))|\bar x|^\beta \,\mathrm{d} t$ with
probabilities $\bar p_\pm=\bar p_\pm(\alpha,C_+/C_-,\beta, A_+/A_-)$, see
[Pilipenko and Proske, Stat. Probab. Lett., 132:62-73, 2018]. In this paper we
solve the generalized selection problem for the stochastic differential
equation $\mathrm{d} X^\varepsilon=a(X^\varepsilon)\,\mathrm{d} t+\varepsilon
b(X^\varepsilon)\,\mathrm{d} Z$ whose dynamics in the vicinity of the origin in
certain sense reminds of dynamics of the model equation. In particular we show
that solutions $X^\varepsilon$ also converge to the maximal or minimal
solutions of the limiting irregular ordinary differential equation $\mathrm{d}
x=a(x) \,\mathrm{d} t$ with the same model selection probabilities $\bar
p_\pm$. This means that for a large class of irregular stochastic differential
equations, the selection dynamics is completely determined by four local
parameters of the drift and the jump measure.
