  Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a
basic classical Lie superalgebra over an algebraically closed field \mathbb{K}
whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be
the reductive algebraic group over \mathbb{K} such that
\mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose
e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the
centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the
centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and
\mathfrak{z}(\mathfrak{g}^{e}) by using associated cocharacters
\tau:\mathbb{K}^{\times}\rightarrow G_{\bar{0}} of e. In addition, we give the
classification of e which are reachable, strongly reachable or satisfy the
Panyushev property for exceptional Lie superalgebras D(2,1;\alpha), G(3) and
F(4).
