The problem (for choosen numerical parameters) is ill-conditionned because it can be reduced to

which has no solution appart from the a trivial one where a__1=0 and c__1=0 (which solve cannot find).
However, the way to solve, that I have provided below, should give solutions for other numerical parameters. Here is an updatad version using exact numbers
nonlin_sys.mw

@GFY

Yes, that is what I meant by: "For equation 3 you get in the same way 2 more equations sets of solutions".

These solutions are only valid for a subset of equations: equation 3, and equation 2, and equation 4.

I did not check if there is in all subsets of solutions a theta_3 that fullfils equation 1 and 3 at the same time.

We are dealing here with a complex argument which is beyond my comfort zone

FunctionAdvisor(plot,tan)

It seems that the real part of the two different solutions for theta__3 is Pi/2 spaced appart.

@acer 

I would not call it a bug. It comes with the design of the labels. Can't the labels be turned on permanetly in help pages. Its essential information when referred to. I use them allot in techincal documentation.

I cannot reprocude the mixed input of 1D Math and 2D Math.

I did in a rather new Maple session (Maple started in 2D):

?D  -> changed to 1D view -> copied the input -> new worksheet -> pasted the input

@dharr 

I will pay attention to this in the next update.

@Axel Vogt 

I could fix it. Thanks

 @rlopez @Carl Love @Ronan 

I had two Maple sessions running for 2 weeks. About a week ago, I noticed that the submenus had reappeared and I could not trigger the disappearance as described above.

Yesterday I restarted the laptop and the first application I started was Maple 2023 followed with a bit delay Maple 2024 (see startup times below). Submenues disappread in both Maple instances, as expected, after the system went to hibernation. After starting a pdf reader (Foxit), the submenus were back.

I could repeatedly toggle submenu dissaperance by: closing Foxit -> Hibernation -> wakeup (&checking if submenus were gone) -> starting Foxit (&checking if submenus were back).

After system wakeup today: No submenues. Toggling as described above did not work. After about an hours submenus were back in the Maple 2023 session. The only activities I can remember were browsing in Firefox, Outlook and web applications running in Microsoft edge.

After about an hour the Submenus appeard again in the Maple2024 session.
These are the tasks running in explorer.exe and their startup times (Maple 2023 java task highlighted in grey)

I have now restarted my other Laptop with one Maple 2024 Session. After the first hibernation submenus disappeard. I could not get the menues back as described above.
After writing this post I checked again and ... the menus are back.
I will try to repeat this. This time without launching any other programs appart from Maple 2024.

@Christian Wolinski 

Chrome uses information from previous user entries (on all web forms not only MaplePrimes) to help you filling in, autocomplete or correct new entries in web forms. Chrome tries to guess what you want to enter. Whether this happens with-in Chrome or accros applications is difficult to say. The tendency is clear. With AI assistance tools all user actions (including old mails and documents) are targeted to be analysed to make our live easier for "free" (there is no free lunch).

I use mostly Firefox and have so far not seen login credentials suggested in searches.

What is wrong here? I cannot find this package.

@GFY 

My initial guess that theta_3 and theta_4 are free to chose was incroeect. I striked it through in teh above. What describes it better: Theta__3 does not depend on theta__4. I.e., for a valid theta_3 there are two valid values for theta__4.

When you plug in the above 16 solutions for a_1 and c_1 into the 4 equations and solve for theta_3 and theta_4, you will get all combinations of variables (in total 64) that are solutions for the system of equations

Here is the first solution for theta__3 from equation 1

subs(a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I, secular7[1]);
solve(%, [theta__3]);
      /               12                 9  \            
      \-1.888144781 10   - 1.978005270 10  I/ ((0.0063172

         - 0.1217592075 I) cos(theta__3)

         + (0.121759209 + 0.006316869337 I) sin(theta__3))


          [[theta__3 = -0.8090474571 + 6.755468571 I]]

which combines with two solutions for theta_4 from equations 2 and 4 to

[a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I],[theta__3 = -0.8090474571 + 6.755468571*I],[theta__4 = -0.7836748989 + 8.209192730*I];
[a__1 = 5.454567591 + 0.005714161093*I, c__1 = 0.002689473223 - 3.665495195*I],[theta__3 = -0.8090474571 + 6.755468571*I],[[theta__4 = 0.7871214279 + 8.209192730*I]]

For equation 3 you get in the same way 2 more equations sets of solutions

@NIMA112 

Here is the idea for H[1] which computes much faster. Carefully check if the expressions were copied correctly. C must be part of the sum since it contains n. Check if that is really what you want. The resutling expression is still big.

ABC.mw

No error with 2023.2

@GFY 

Below is a way to solve for the 16 roots. As said above theta_3 and theta_4 are free to choose. If you replace the floats by parameters or rational numbers, this solution will be exact.
 

Download solve821_reply.mw

Have you checked for backups?
Depending on how Maple is terminated there should be backups if the options are configured like this

I thought when odetest returns zero without assumptions then the tested solution is valid over the complex domain. In your example this is not the case.

The worring consequence is that plotting for a given parameter the solution of one of the "root ode"s (where the residual of the identical solution of the origianl ode is non zero) does not show an empty plot. For this reason, I am now also of the opinion that one should work with the odetest of the original ode.  There is a flaw in my reasoning that no assumptions are made when using PDEtools:-Solve that I would like to understand.