What about Markiyan Hirnyk  and MaplePrimes Search?

Directly by using Maple functions dsolve/numeric/BVP
managed to find only one solution for many initial data ('approxsoln').
If anyone, of course, is interesting.
 

restart; 
Digits := 20:
dsys := {diff(x(t), t, t)+.2*(diff(x(t), t))+x(t)^3-.3*cos(t) = 0, x(0) = x(2*Pi), (D(x))(0) = (D(x))(2*Pi)}:
sol := dsolve(dsys, numeric, abserr = 1.*10^(-5), approxsoln = [x(t) = .5, (D(x))(t) = 1.5]):
plots[odeplot](sol, [diff(x(t), t), x(t)], 0 .. 2*Pi)

@Janeasefor Thank you very much, very pleased  that you liked it.

In principle, I think it is possible to reduce all “inverse kinematics” to one subroutine (for example, in Maple) for any kind of manipulators in order to obtain a solution based on a straight line segment. More precisely, I do not think, in fact, everything is checked in Maple.
 

@vv 
I do not mind in any way. I have long been interested in finding out whether dsolve works after the “event”.  And it was appeared a reason to do it.

@rlopez 
The word "correct" does not quite match the situation. It’s just that Anatoly Vladimirovich Draghilev himself believed that the spelling of his last name in English “Draghilev” corresponded to reading in Russian, well, he asked for it.
Once again, thank you for your attention to this issue. It seems to me that the main thing is that you did a very good job.

@verdin 
In the neighborhood of a solution point there will always be a continuous solution (according to the theory). But constraints on variables will have to be chosen from the found set of solutions.
Draghilev's method for your situation is well stated here
https://www.maplesoft.com/applications/view.aspx?SID=149514
If translation from Russian is available to you, then
https://vk.com/doc242471809_437831729

@verdin 
If there are continuous solutions of the system of m underdetermined equations (m*n, n>m), it is possible to obtain the dependence of all variables: xi, i = 1..n (where "a" is one of xi) from a certain parameter. For example, the  Draghilev's method. It is suitable?

@verdin 
I'm afraid you misunderstood. In its original form, your system most likely has no solutions.
It was shown that the separately numerator and denominator (that after "/") have the same solutions. These were equations without denominators in fsolve.

@verdin 
Your system of equations seems to have dependencies and features. In the first text I removed the denominators and found a solution with the help of fsolve. The same values of variables give 0 of denominators (second text).
Please check carefully.
Alg_equation_numerators.mw
Alg_equation_denominators.mw

@verdin 
We are not talking about the Draghilev method, we are talking directly about your system of equations, which needs to be reduced in order to try to get at least some solution - just a point.
I simply showed a way to reduce the system at the expense of the very simple expression of some variables through other variables and the removal of denominators. ( I did not invent anything, but I started with your equations: f1,f2,f3,f4)

@verdin 

You first need to simplify your system of equations. You will have, as a maximum, 5 variables (x4, x5, x6, x7, a) and four equations f5, f6, f7, f8.
Here, for example, the first steps.
Alg_equation_trans_f5.mw

You also need to remove the denominators from the equations, but when you find the solutions, you must to check these solutions to equality 0 denominators.

@rlopez 
Thank you for your attention to the spelling of the last name Draghilev.

Examples:  2x1^2 + x2^2 + x3^2 - 0.5^2 = 0; and x1^4 + x2^4 + x3^4 - 0.5^4 = 0;

 

The algorithm is still being finalized.

As for me, then no, I cannot.  But I can promise, if something turns out interesting, I will definitely try  implement in Maple.