  Traveling wave solutions of degenerate coupled $\ell$-KdV equations are
studied. Due to symmetry reduction these equations reduce to one ODE,
$(f')^2=P_n(f)$ where $P_n(f)$ is a polynomial function of $f$ of degree
$n=\ell+2$, where $\ell \geq 3$ in this work. Here $\ell$ is the number of
coupled fields. There is no known method to solve such ordinary differential
equations when $\ell \geq 3$. For this purpose, we introduce two different type
of methods to solve the reduced equation and apply these methods to degenerate
three-coupled KdV equation. One of the methods uses the Chebyshev's Theorem. In
this case we find several solutions some of which may correspond to solitary
waves. The second method is a kind of factorizing the polynomial $P_n(f)$ as a
product of lower degree polynomials. Each part of this product is assumed to
satisfy different ODEs.
