Let $\alpha_{j}(z)$, $j=1,2$, $a_{i}(z)$, $i=1,2,\ldots,6$ be meromorphic functions. Suppose the differential equation (*) $w'^3+\alpha_2(z)w'^2+\alpha_1(z)w'=a_6(z)w^6+\cdots+a_1(z)w+a_0(z)$ possesses an admissible solution $w(z)$. If $\eta(z)$ is a solution of (*) and small with respect to $w(z)$ and if (*) is irreducible, then $\eta(z)$ is a deficient or a ramified function for $w(z)$.

We give a bijective lattice path proof of the equality of the dual Jacobi-Trudy determinant formulas for Schur polynomials. Related ideas have appeared in [1, pp.304-306] and [2, p.24]. We remark that the same bijection works for the case of flagged skew Schur polynomials [2, 8] and that a determinant for $q$-counting restricted lattice paths [7] follows from the bijection.

The main purpose of this paper is to classify integral surfaces of the unit sphere $S^5(1)$ which are mass-symmetric and of 2-type. If we consider $S^5(1)$ as a Sasakian manifold, then we prove that a mass-symmetric 2-type integral surface of $S^5(1)$ lies fully in $S^5(1)$ and is the product of a plane circle and a helix of order 4 or the product of two circles.