  Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and
a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced
$\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 0})} F $ is called
a finite Verma module. In the present paper we classify the non-irreducible
finite Verma modules over the largest exceptional linearly compact Lie
superalgebra ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ of
minimal codimension. This is done via classification of all singular vectors in
the modules $M(F)$. Besides known singular vectors of degree 1,2,3,4 and 5, we
discover two new singular vectors, of degrees 7 and 11. We show that the
corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be
arranged in an infinite number of bilateral infinite complexes, which may be
viewed as 'exceptional' de Rham complexes for $E(5,10)$.
