How to solve this system of coupled partial differential equations

I am trying to solve the following system analytically but apparently Mathematica does not know how to do it (although I know there exists an analytic solution). Can anyone help me?

Thank you!!

There are 4 functions of 4 variables to determine with the following 16 coupled equations:

  D[pr[t, x, r, th], t]*t + pr[t, x, r, th] == 0, 
  D[px[t, x, r, th], t]*t + px[t, x, r, th] == 0, 
  D[pth[t, x, r, th], t]*t + pth[t, x, r, th] == 0, 
  D[pt[t, x, r, th], r] == 0, 
  D[pr[t, x, r, th], r]*t + pt[t, x, r, th] == 
   0, (2 + k r)*D[px[t, x, r, th], r] + k px[t, x, r, th] == 
   0, (2 + k r)*D[pth[t, x, r, th], r] + k pth[t, x, r, th] == 
   0, (4 + k x^2)^2 D[pt[t, x, r, th], x] - 
    4 k (2 + kr) t px[t, x, r, th] == 
   0, (4 + k r^2)^2 D[pr[t, x, r, th], x] - 
    4 (-4 + k^2 r^2) px[t, x, r, th] == 0, 
  D[t (2 + k r) (4 + k x^2) px[t, x, r, th], 
     x] + (2 + k r) (4 + k x^2) pt[t, x, r, th] + 
    t k (4 + kx^2) pr[t, x, r, th] - 
    2 k x t (2 + k r) px[t, x, r, th] == 0, 
  D[(4 x + k x^2) pth[t, x, r, th], 
     x] + (4 - k x^2) pth[t, x, r, th] == 
   0, (4 + k x^2)^2 D[pt[t, x, r, th], th] - 
    4 k (2 + k r) t x^2 pth[t, x, r, th] == 
   0, (4 + k x^2)^2 D[pr[t, x, r, th], th] - 
    4 (k^2 r^2 - 4) x^2 pth[t, x, r, th] == 
   0, (4 + k x^2) D[px[t, x, r, th], th] - (-4 + k x^2) x pth[t, x, r,
       th] == 0, 
  t (2 + k r) (4 x + k x^3) D[pth[t, x, r, th], 
      th] + (4 - k x^2) (2 + k r) t px[t, x, r, 
      th] + (2 + k r) (4 x + k x^3) pt[t, x, r, th] + 
    k t (4 - k x^2) prt (2 + k r) (4 x + k x^3) == 0}, {pt[t, x, r, 
   th], px[t, x, r, th], pr[t, x, r, th], pth[t, x, r, th]}, {t, x, r,
   th}]