  Singular vectors of a representation of a finite-dimensional simple Lie
algebra are weight vectors in the underlying module that are nullified by
positive root vectors. In this article, we use partial differential equations
to find all the singular vectors of the polynomial representation of the simple
Lie algebra of type $F_4$ over its basic irreducible module. As applications,
we obtain a new combinatorial identity about the number 24 and explicit
generators of invariants. Moreover, we show that the number of irreducible
submodules contained in the space of homogeneous harmonic polynomials with
degree $k\geq 2$ is $\geq [|k/3|]+[|(k-2)/3|]+2$.
