A nonlinear system of 1st order ODEs

Question

I have the following nonlinear system of 1st order ODEs:

$e^p\dfrac{dy}{dp}=k(\cosh{w})^{2n}$
and
$\dfrac{1}{y+1}\left(\dfrac{dy}{dp}+y\right)=\left(2(n+1)\dfrac{dw}{dp}\right)^2$
where $k$ is a non-zero parameter and $1<n\leq3$ is a rational number. This has a trivial solution $y(p)=0, w(p)=\cosh^{-1}{0}$ for all values of n. I would like to know:

1. Is there any way to obtain a non-trivial particular solution of this system for arbitrary values of n?
2. There is a non-trivial solution for $n=3$:
   $y(w)=\dfrac{1}{9}\left(\dfrac{2}{(\sinh{w})^4}-5\right)^2-1$
   and
   $p(w)=\ln\left(\dfrac{9k(\sinh{w})^8(\cosh{w})^4}{4(2(\sinh{w})^2-1)}\right)$
   Are there other such particular solutions for specific values of $n$?

Any input is greatly appreciated. Thanks.