These are initial conditions for the members of the classes of orbits I have found.

For each angle, I start with a guess and dyadically approximate the value of v0
c-----if previous v0 overshoots
c     v0 = guess+((2n-1).d0)/((2n).d0)
c-----if previous v0 undershoots
c     v0 = guess+((2n+1).d0)/((2n).d0)


c------[Figure 8](2 lobe)
      theta = pi/4.d0
c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /
c------overshoots
c     v0 = 27.d0/32.d0
c     v0 = 183.d0/256.d0
c     v0 = 365.d0/512.d0
c     v0 = 729.d0/1024.d0
c     v0 = 23327.d0/32768.d0
c------/overshoots

      v0 = 46653.d0/65536.d0

c------undershoots
c     v0 = 46653.d0/65536.d0
c     v0 = 11663.d0/16384.d0
c     v0 = 5831.d0/8192.d0
c     v0 = 2915.d0/4096.d0
c     v0 = 1457.d0/2048.d0
c     v0 = 91.d0/128.d0
c     v0 = sqrt(1.d0/2.d0)
c     v0 = 181.d0/256.d0
c------/undershoots

c------[Six Lobe -x-]
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 7.d0*pi/15.d0
c------overshoots
c     v0 = 1.d0+2727.d0/8192.d0
c     v0 = 1.d0+5453.d0/16384.d0
c     v0 = 1.d0+341.d0/1024.d0
c     v0 = 1.d0+171.d0/512.d0
c     v0 = 1.d0+10905.d0/32768.d0
c------/overshoots

       v0 = 1.d0+21809.d0/65536.d0
c------undershoots
c     v0 = 1.d0+1363.d0/4096.d0
c     v0 = 1.d0+681.d0/2048.d0
c     v0 = 4.d0/3.d0
c------/undershoots


dt
.0075
numpoints
6000

c------[Six Lobe -X-]
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 10.d0*pi/21.d0
c------overshoots
c     v0 = 2.d0
c     v0 = 1.d0+(1.d0)/(2.d0)
c     v0 = 1.d0+(3.d0)/(8.d0)
c     v0 = 1.d0+(5.d0)/(16.d0)
c     v0 = 1.d0+(19.d0)/(64.d0)
c     v0 = 1.d0+(37.d0)/(128.d0)
c     v0 = 1.d0+(73.d0)/(256.d0)
c     v0 = 1.d0+(145.d0)/(512.d0)
c     v0 = 1.d0+(289.d0)/(1024.d0)
c     v0 = 1.d0+(4623.d0)/(16384.d0)
c------/overshoots
      v0 = 1.d0+(18491.d0)/(65536.d0)
c     v0 = 84027/65536;
c------undershoots
c     v0 = 1.d0+(9245.d0)/(32768.d0)
c     v0 = 1.d0+(2311.d0)/(8192.d0)
c     v0 = 1.d0+(1155.d0)/(4096.d0)
c     v0 = 1.d0+(577.d0)/(2048.d0)
c     v0 = 1.d0+(9.d0)/(32.d0)
c     v0 = 1.d0+(1.d0)/(4.d0)
c     v0 = 1.d0
c------/undershoots
  //[Six Lobe -X-]
  theta = 10.d0*pi/21.d0
  v0 = 1.0+(18491.0)/(65536.0)

10 lobe orbit
If you use too big of a time step for this orbit, it will drift vertically.
I got the same value for the initial velocity using two separate dyadic approximations.
That is 1.3 + 2713/163840 is the same as 1 + 10373/32768 which both are
43141/32768
Therefore this orbit *may* have a RATIONAL initial velocity, which to me is quite surprising.
The only way to be sure is to prove that it is a solution to the differential equation. The only
problem there is finding a suitable form of this solution.
I have checked with a very small step size (about 1/1000th the length I normally use) and the lines
still seem to be within the margins of error of the program.
      theta = 52.d0*pi/105.d0
c     v0 = 1.325d0
c     v0 = 1.3125d0
c     v0 = 1.31875d0
c     v0 = 1.3d0+(3.d0)/(160.d0)
c     v0 = 1.3d0+(5.d0)/(320.d0)
c     v0 = 1.3d0+(11.d0)/(640.d0)
c     v0 = 1.3d0+(21.d0)/(1280.d0)
c     v0 = 1.3d0+(43.d0)/(2560.d0)
c     v0 = 1.3d0+(85.d0)/(5120.d0)
c     v0 = 1.3d0+(169.d0)/(10240.d0)
c     v0 = 1.3d0+(339.d0)/(20480.d0)
c     v0 = 1.3d0+(679.d0)/(40960.d0)
c     v0 = 1.3d0+(1357.d0)/(81920.d0)
      v0 = 1.3d0+(2713.d0)/(163840.d0)

c------[Ten Lobe ->X<-]
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 52.d0*pi/105.d0
c------overshoots
c     v0 = 1.d0 + (1.d0)/(2.d0)
c     v0 = 1.d0 + (3.d0)/(8.d0)
c     v0 = 1.d0 + (11.d0)/(32.d0)
c     v0 = 1.d0 + (21.d0)/(64.d0)
c     v0 = 1.d0 + (41.d0)/(128.d0)
c     v0 = 1.d0 + (163.d0)/(512.d0)
c     v0 = 1.d0 + (325.d0)/(1024.d0)
c     v0 = 1.d0 + (649.d0)/(2048.d0)
c     v0 = 1.d0 + (1297.d0)/(4096.d0)
c     v0 = 1.d0 + (5187.d0)/(16384.d0)
c     v0 = 1.d0 + (10373.d0)/(32768.d0)
c------/overshoots

c------undershoots
c     v0 = 1.d0 + (41491.d0)/(131072.d0)
c     v0 = 1.d0 + (20745.d0)/(65536.d0)
c     v0 = 1.d0 + (2593.d0)/(8192.d0)
c     v0 = 1.d0 + (81.d0)/(256.d0)
c     v0 = 1.d0 + (9.d0)/(32.d0)
c     v0 = 1.d0 + (5.d0)/(16.d0)
c     v0 = 1.d0 + (1.d0)/(4.d0)
c     v0 = 1.d0
c------/undershoots

c------[Fourteen Lobe Orbit ->)][(<-]
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 472.d0*pi/945.d0
c------overshoots
c     v0 = 2.d0
c     v0 = 1.d0+(1.d0)/(2.d0)
c     v0 = 1.d0+(3.d0)/(8.d0)
c     v0 = 1.d0+(5.d0)/(16.d0)
c     v0 = 1.d0+(39.d0)/(128.d0)
c     v0 = 1.d0+(155.d0)/(512.d0)
c     v0 = 1.d0+(1239.d0)/(4096.d0)
c     v0 = 1.d0+(2477.d0)/(8192.d0)
c     v0 = 1.d0+(4953.d0)/(16384.d0)
c------/overshoots

      v0 = 1.d0+(19811.d0)/(65536.d0)

c------undershoots
c     v0 = 1.d0+(9905.d0)/(32768.d0)
c     v0 = 1.d0+(619.d0)/(2048.d0)
c     v0 = 1.d0+(309.d0)/(1024.d0)
c     v0 = 1.d0+(77.d0)/(256.d0)
c     v0 = 1.d0+(19.d0)/(64.d0)
c     v0 = 1.d0+(9.d0)/(32.d0)
c     v0 = 1.d0+(1.d0)/(4.d0)
c     v0 = 1.d0
c------/undershoots

c------[Fourteen Lobe Orbit ->)}{(<-]
c-----This orbit is interesting, it has 14 lobes just as the one above does.
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 577.d0*pi/1155.d0
c------overshoots
c     v0 = 1.d0 + (1.d0)/(2.d0)
c     v0 = 1.d0 + (3.d0)/(8.d0)
c     v0 = 1.d0 + (5.d0)/(16.d0)
c     v0 = 1.d0 + (151.d0)/(512.d0)
c     v0 = 1.d0 + (301.d0)/(1024.d0)
c     v0 = 1.d0 + (601.d0)/(2048.d0)
c     v0 = 1.d0 + (2403.d0)/(8192.d0)
c     v0 = 1.d0 + (9611.d0)/(32768.d0)
c     v0 = 1.d0 + (19221.d0)/(65536.d0)
c     v0 = 1.d0 + (38441.d0)/(131072.d0)
c------/overshoots

c     v0 = 1.d0 + (76881.d0)/(262144.d0)

c------undershoots
c     v0 = 1.d0 + (76881.d0)/(262144.d0)
c     v0 = 1.d0 + (4805.d0)/(16384.d0)
c     v0 = 1.d0 + (1201.d0)/(4096.d0)
c     v0 = 1.d0 + (75.d0)/(256.d0)
c     v0 = 1.d0 + (37.d0)/(128.d0)
c     v0 = 1.d0 + (19.d0)/(64.d0)
c     v0 = 1.d0 + (9.d0)/(32.d0)
c     v0 = 1.d0 + (1.d0)/(4.d0)
c     v0 = 1.d0
c------/undershoots


c--other orbit shapes
c------[3pi/7]
c-----This orbit is close to a stable orbit, but it is very sensitive to dt
c-----and the initial conditions, probably because it approaches the masses
c-----so closely.
      theta = 3.d0*pi/7.d0
c------overshoots
c     v0 = 1.d0 + (1.d0)/(2.d0)
c     v0 = 1.d0
c     v0 = (15.d0)/(16.d0)
c------/overshoots
      v0 = (59.d0)/(64.d0)
      v0 = (119.d0)/(128.d0)
      v0 = (117.d0)/(128.d0)
c------undershoots
c     v0 = (29.d0)/(32.d0)
c     v0 = (7.d0)/(8.d0)
c------/undershoots

c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /

c------[Figure 8 4pi/13]
c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /
      theta = 4.d0*pi/13.d0
c------overshoots
1
1023/1024
4091/4096
16363/16384
130903/131072
c------/undershoots

c------[36pi/89]
This orbit I found (2006-10-27) after generating 2 random integers between 0 and 100 and checking if there was an orbit for this angle.
The integers generated were 53 and 89, so I changed the fraction so that it was pointing into the first quadrant (36pi/89).
I then, rather quickly, found this solution. Is there a closed orbit for all rational multiples of 2pi? I think so.
c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
      theta = 36.d0*pi/89.d0
c------overshoots
c     v0 = 1.d0 + (1.d0)/(4.d0)
c     v0 = 1.d0 + (3.d0)/(16.d0)
c     v0 = 1.d0 + (1.d0)/(8.d0)
c     v0 = 1.d0 + (15.d0)/(128.d0)
c     v0 = 1.d0 + (29.d0)/(256.d0)
c     v0 = 1.d0 + (57.d0)/(512.d0)
c------/overshoots
      v0 = 1.d0 + (14463.d0)/(131072.d0)
c     v0 = 145535/131072;
c     dt = .0001;
c     numpoints = 150000;
c------undershoots
c     v0 = 1.d0 + (7231.d0)/(65536.d0)
c     v0 = 1.d0 + (3615.d0)/(32768.d0)
c     v0 = 1.d0 + (1807.d0)/(16384.d0)
c     v0 = 1.d0 + (903.d0)/(8192.d0)
c     v0 = 1.d0 + (451.d0)/(4096.d0)
c     v0 = 1.d0 + (225.d0)/(2048.d0)
c     v0 = 1.d0 + (113.d0)/(1024.d0)
c     v0 = 1.d0 + (7.d0)/(64.d0)
c     v0 = 1.d0 + (3.d0)/(32.d0)
c     v0 = 1.d0 + (1.d0)/(16.d0)
c     v0 = 1.d0
c     v0 = (7.d0)/(8.d0)
c     v0 = (3.d0)/(4.d0)
c------/undershoots
  //[36pi/89]
  theta = 36.0*pi/89.0;
  v0 = 1.0 + (14463.0)/(131072.0);

c------[18pi/89]
c-----For this orbit  // is an undershoot and // is an overshoot
cx                    /                     /
      theta = 18.d0*pi/89.d0
c     dt = .00002;
c     numpoints = 1200000;
c------overshoots
c     v0 = 157.d0/128.d0
c     v0 = 623.d0/512.d0
c     v0 = 1243.d0/1024.d0
c------/overshoots
c------undershoots
c     v0 = 621.d0/512.d0
c     v0 = 607.d0/512.d0
c     v0 = 303.d0/256.d0
c     v0 = 151.d0/128.d0
c     v0 = 150.d0/128.d0
c------/undershoots


c------[Conductor Orbit]
      theta = pi/7.d0
c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /
c------overshoots
c     v0 = 1.d0 + (25.d0)/(512.d0)
c     v0 = 1.d0 + (47.d0)/(1024.d0)
c     v0 = 1.d0 + (187.d0)/(4096.d0)
c     v0 = 1.d0 + (747.d0)/(16384.d0)
c     v0 = 1.d0 + (747.d0)/(16384.d0)
c------/overshoots
      v0 = 1.d0 + (2987.d0)/(65536.d0)
c------undershoots
c     v0 = 1.d0 + (1493.d0)/(32768.d0)
c     v0 = 1.d0 + (373.d0)/(8192.d0)
c     v0 = 1.d0 + (93.d0)/(2048.d0)
c     v0 = 1.045d0
c     v0 = 1.d0 + (23.d0)/(512.d0)
c     v0 = 1.d0 + (21.d0)/(512.d0)
c------/undershoots

c------[Flower Orbit]
      theta = 1.d0*pi/5.d0
c------overshoots
c     v0 = 1.d0 + (1.d0)/(2.d0)
c     v0 = 1.d0 + (1.d0)/(4.d0)
c     v0 = dsqrt(dsqrt(2.d0))
c     v0 = 1.d0 + (3.d0)/(16.d0)
c------/overshoots

      v0 = 1.d0 + (23.d0)/(128.d0)

c------undershoots
c     v0 = 1.d0 + (11.d0)/(64.d0)
c     v0 = 1.d0 + (1.d0)/(8.d0)
c     v0 = 1.d0
c------/undershoots

c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /
c------[Flower Orbit]
      theta = 4.d0*pi/17.d0
c------overshoots
c     v0 = (17.d0)/(16.d0)
c     v0 = (33.d0)/(32.d0)
c     v0 = (131.d0)/(128.d0)
c     v0 = (263.d0)/(256.d0)
c     v0 = (2103.d0)/(2048.d0)
c     v0 = (16823.d0)/(16384.d0)
c     v0 = (8411.d0)/(8192.d0)
c------/overshoots

c     v0 = (33643.d0)/(32768.d0)

c------undershoots
c     v0 = (16823.d0)/(16384.d0)
c------/undershoots

c-----For this orbit  // is an overshoot and // is an undershoot
c                     /                     /
c------[2pi/5]
c     theta = 2.d0*pi/5.d0

c------overshoots
c     v0 = (17.d0)/(16.d0)
c     v0 = (1589.d0)/(2048.d0)
c     v0 = (25421.d0)/(32768.d0)
c------/overshoots

c     v0 = (50841.d0)/(65536.d0)

c------undershoots
c     v0 = (6355.d0)/(8192.d0)
c     v0 = (199.d0)/(256.d0)
c     v0 = (99.d0)/(128.d0)
c     v0 = (25.d0)/(32.d0)
c     v0 = (13.d0)/(16.d0)
c------/undershoots
c     2pi/5
c     199/256
c     timestep: .00001
c     iterations: 1000000

c-----For this orbit  // is an overshoot and // is an undershoot
c                    /                       /
c------[2pi/5]
c     theta = 2.d0*pi/5.d0

c------overshoots
c     v0 = (77.d0)/(64.d0)
c------/overshoots

c     v0 = (307.d0)/(256.d0)

c------undershoots
c     v0 = (153.d0)/(128.d0)
c     v0 = (19.d0)/(16.d0)
c------/undershoots
c     2pi/5
c     19/16
c     timestep: .00001
c     iterations: 1000000

c--old values

c------[]
c     theta = pi/5.d0
c     v0 = dsqrt(3.d0)/(3.d0)
c     v0 = 26.d0/45.d0

c     v0 = dsqrt(dsqrt(2.d0))

c------[]
c     theta = pi/5.d0
c     theta = (7.d0)*pi/(36.d0)
c     theta = pi/3.d0
c     v0 = dsqrt(.5d0)
c     v0 = dsqrt(3.d0)/1.68033d0

Guesses

7/15                1 21809/65536       87345
52/105              1 20746/65536       86282           (1063)      (.98782987)
472/945             1 19811/65536       85347           (935)       (.989163441)
577/1155           [1 19000/65536       84536           (811)       (.990498812)]


unfinished orbits
shape-ish	theta		starting point
asterisk	5pi/13		43/32
six-lobe-ii	8pi/17		21/16, 673/512, 2693/2048
figure-8-iii	5pi/17		57/64
3/7		3pi/7		117/128-118/128

for fun
v0	theta
7/8	5/16, 41/128